Geometric algebra of strictly convex Minkowski planes

Geometric algebra of strictly convex Minkowski
planes ∗
Margarita Spirova
TU Chemnitz
Emerging Developments in Real Algebraic Geometry
February 20-24, 2012, Magdeburg
∗ Based on joint work with Horst Martini and Karl Strambach
1 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Motivation
2 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
Let (M d ,·) be a Minkowski space, i.e., a finite dimensional real
vector space equipped with a norm ·.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
Let (M d ,·) be a Minkowski space, i.e., a finite dimensional real
vector space equipped with a norm ·.
An isometry from (X,·X) to (Y,·Y) is a mapping such that
for all x1,x2 in X we have
x1 −x2X = Tx1 −Tx2Y.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
• Translations, the identity map I, and the symmetry −I are
isometries for any Minkowski space.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
• Translations, the identity map I, and the symmetry −I are
isometries for any Minkowski space.
• Mazur-Ulam Theorem: If (X,·X) to (Y,·Y) are
Minkowski spaces and T is an isometry of (X,·X) into
(Y,·Y) with T(0) = 0 then T is linear.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
• Translations, the identity map I, and the symmetry −I are
isometries for any Minkowski space.
• Mazur-Ulam Theorem: If (X,·X) to (Y,·Y) are
Minkowski spaces and T is an isometry of (X,·X) into
(Y,·Y) with T(0) = 0 then T is linear.
• Álvarez Paiva-Thompson Remark: A general Minkowski
plane does not admit many isometries itself.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski spaces
• Translations, the identity map I, and the symmetry −I are
isometries for any Minkowski space.
• Mazur-Ulam Theorem: If (X,·X) to (Y,·Y) are
Minkowski spaces and T is an isometry of (X,·X) into
(Y,·Y) with T(0) = 0 then T is linear.
• Álvarez Paiva-Thompson Remark: A general Minkowski
plane does not admit many isometries itself.
⎧
⎨
⎩
x = (ξ1,ξ2) :=
|ξ1|+2 −1 |ξ2| if sgnξ1 = sgnξ1 and |ξ2| ≤ 2|ξ1|,
|ξ2| if sgnξ1 = sgnξ1 and |ξ2| > 2|ξ1|,
(ξ2 1 +ξ2 2 )12
if sgnξ1 = −sgnξ1.
3 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
• When an isometry Φ is involutory it always has fixed points.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
• When an isometry Φ is involutory it always has fixed points.If
Φ has a line G of fixed points than it is called a reflections in
G.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
• When an isometry Φ is involutory it always has fixed points.If
Φ has a line G of fixed points than it is called a reflections in
G.
• If any line in a Minkowski plane M 2 admits a reflection then
M 2 is Euclidean.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
• When an isometry Φ is involutory it always has fixed points.If
Φ has a line G of fixed points than it is called a reflections in
G.
• If any line in a Minkowski plane M 2 admits a reflection then
M 2 is Euclidean.
H. Martini, M. S.: Reflections in strictly convex Minkowski
planes, Aequationes Math. 78 (2009), 71-85.
4 / 24

Motivation
Isometries in
Minkowski spaces
Reflections in
Minkowski planes
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Reflections in Minkowski planes
H. Busemann, P. J. Kelly: Projective Geometry and
Projective metrics, New York, 1953.
• An isometry of a Minkowski plane has either no fixed points,
one fixed point, a line of fixed points, or it is identity.
• When an isometry Φ is involutory it always has fixed points.If
Φ has a line G of fixed points than it is called a reflections in
G.
• If any line in a Minkowski plane M 2 admits a reflection then
M 2 is Euclidean.
H. Martini, M. S.: Reflections in strictly convex Minkowski
planes, Aequationes Math. 78 (2009), 71-85.
Bachmann, F.: Aufbau der Geometrie aus dem
Spiegelungsbegriff, Springer-Verlag, Berlin-Göttingen-Heidelberg,
1959.
4 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Preliminares
5 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Let (M d ,·) be a Minkowski space.
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
A non-zero vector x ∈ M d is Birkhoff orthogonal to a non-zero
vector y ∈ M d , denoted by x ⊥B y, if
for all λ ∈ R.
x ≤ x+λy
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
A non-zero vector x ∈ M d is Birkhoff orthogonal to a non-zero
vector y ∈ M d , denoted by x ⊥B y, if
for all λ ∈ R.
x ≤ x+λy
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
A non-zero vector x ∈ M d is Birkhoff orthogonal to a non-zero
vector y ∈ M d , denoted by x ⊥B y, if
for all λ ∈ R.
x ≤ x+λy
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
A non-zero vector x ∈ M d is Birkhoff orthogonal to a non-zero
vector y ∈ M d , denoted by x ⊥B y, if
for all λ ∈ R.
x ≤ x+λy
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
Birkhoff orthogonality
A non-zero vector x ∈ M d is Birkhoff orthogonal to a non-zero
vector y ∈ M d , denoted by x ⊥B y, if
for all λ ∈ R.
x ≤ x+λy
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
James orthogonality
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
James orthogonality
A non-zero vector x ∈ M d is said to be (isosceles or) James
orthogonal to a non-zero vector y, denoted by x ⊥J y, if
x+y = x−y.
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
James orthogonality
A non-zero vector x ∈ M d is said to be (isosceles or) James
orthogonal to a non-zero vector y, denoted by x ⊥J y, if
x+y = x−y.
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
James orthogonality
A non-zero vector x ∈ M d is said to be (isosceles or) James
orthogonal to a non-zero vector y, denoted by x ⊥J y, if
x+y = x−y.
6 / 24

Motivation
Preliminares
Different types of
orthogonality in
Minkowski spaces
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Different types of orthogonality in Minkowski spaces
James orthogonality
A non-zero vector x ∈ M d is said to be (isosceles or) James
orthogonal to a non-zero vector y, denoted by x ⊥J y, if
x+y = x−y.
6 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski planes and
reflections in the group of affine
transformations
7 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski planes
8 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Isometries in Minkowski planes
Theorem: If (M 2 ,·) is a Minkowski plane which is
non-Euclidean, then the group S of isometries of (M 2 ,·) is the
semi-direct product of the translation group T of R 2 with a finite
group which is either a cyclic group of rotations or the dihedral
group D to 2n with 2 ≤ n < ∞ generated by two different affine
reflections.
8 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
A point reflection in the stabilizer of the isometry group
9 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
A point reflection in the stabilizer of the isometry group
Proposition: Let α be the reflection at the point 0 in the
stabilizer S0 of the isometry group S of a strictly convex
Minkowski plane (M 2 ,·). If S0 is a dihedral group, then the
lines Gτ and Gτα, which are axes of the affine reflections τ resp.
τα, are mutually orthogonal.
9 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Affine reflections
10 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Affine reflections
All affine reflections with the axis through 0 and (cost,sint) are
given by the set M of matrices of the form
1 cos2t− 2bsin2t sin2t+bcos2 t
sin2t−bsin 2t −cos2t+ 1
2bsin2t
.
10 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Isometries in
Minkowski planes
A point reflection in
the stabilizer of the
isometry group
Affine reflections
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Affine reflections
All affine reflections with the axis through 0 and (cost,sint) are
given by the set M of matrices of the form
1 cos2t− 2bsin2t sin2t+bcos2 t
sin2t−bsin 2t −cos2t+ 1
2bsin2t
.
Lemma: Let ΨG1 and ΨG2 be two affine reflections along the
non-parallel lines G1 and G2.
(a) The reflections ΨG1 and ΨG2 leave a line H through G1 ∩G2
invariant if and only if their product is a shear with H as axis.
(b) The product ΨG1 ◦ΨG2 has only the intersection p of G1 and
G2 as a fixed point if and only if ΨG1 ◦ΨG2 is not a shear.
10 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Left-reflections in Minkowski planes
Cycles in strictly
convex Minkowski
planes 11 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Definition
Definition: Let there be given a line G in a strictly convex
Minkowski plane (M 2 ,·). A transformation
ΨG : M 2 → M 2
is called a left-reflection along the line G if
(i) p ΨG := p for any p ∈ G;
(ii) p ΨG := p ′
such that 〈p,p ′ 〉 ⊥B G and the midpoint of [p,p ′ ] lies on G if
p ∈ G.
Cycles in strictly
convex Minkowski
planes 12 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
(i) ΨG is involutory.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
(i) ΨG is involutory.
(ii) ΨG is affine.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
(i) ΨG is involutory.
(ii) ΨG is affine.
(iii) If H is an arbitrary line and H ΨG = H ′ , then either
H H ′ G or H ∩H ′ ∈ G.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
(i) ΨG is involutory.
(ii) ΨG is affine.
(iii) If H is an arbitrary line and H ΨG = H ′ , then either
H H ′ G or H ∩H ′ ∈ G.
(iv) The only fixed lines of ΨG except for G are those which are
Birkhoff orthogonal to G, i.e., the lines Birkhoff orthogonal of
G form the direction of ΨG.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Basic properties
Theorem: Let ΨG be a left-reflection in a strictly convex
Minkowski plane (M 2 ,·). Then ΨG has the following properties.
(i) ΨG is involutory.
(ii) ΨG is affine.
(iii) If H is an arbitrary line and H ΨG = H ′ , then either
H H ′ G or H ∩H ′ ∈ G.
(iv) The only fixed lines of ΨG except for G are those which are
Birkhoff orthogonal to G, i.e., the lines Birkhoff orthogonal of
G form the direction of ΨG.
(v) The product of three left-reflections along parallel lines is a
left-reflection along another line belonging to the same pencil
of parallel lines.
Cycles in strictly
convex Minkowski
planes 13 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Relations between point reflections and left-reflection
Cycles in strictly
convex Minkowski
planes 14 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Relations between point reflections and left-reflection
Proposition: Let p be an arbitrary point in a strictly convex
Minkowski plane (M 2 ,·). For the symmetry Υp with respect to
any point p there exist two mutually Birkhoff orthogonal lines G1
and G2 through p such that
Υp = ΨG2
◦ΨG1 .
Cycles in strictly
convex Minkowski
planes 14 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Relations between point reflections and left-reflection
Proposition: Let p be an arbitrary point in a strictly convex
Minkowski plane (M 2 ,·). For the symmetry Υp with respect to
any point p there exist two mutually Birkhoff orthogonal lines G1
and G2 through p such that
Υp = ΨG2
◦ΨG1 .
Lemma: Let G1, G2 be two intersecting lines in a strictly convex
Minkowski plane, and ΨG1 , ΨG2 be the left reflections along these
lines. Then the following statements are equivalent.
(i) G1 and G2 are mutually Birkhoff orthogonal;
(ii) ΨG2 ◦ΨG1 = ΨG1 ◦ΨG2 ;
(iii) ΨG1 ◦ΨG2 is the symmetry with respect to the intersection
point of G1 and G2.
Cycles in strictly
convex Minkowski
planes 14 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Characterizations of special classes of norms via
left-reflections
Cycles in strictly
convex Minkowski
planes 15 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Characterizations of special classes of norms via
left-reflections
Proposition: If every left-reflection in a strictly convex, smooth
Minkowski plane (M 2 ,·) preserves Birkhoff orthogonality, then
(M 2 ,·) is a Radon plane.
Cycles in strictly
convex Minkowski
planes 15 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Characterizations of special classes of norms via
left-reflections
Proposition: If every left-reflection in a strictly convex, smooth
Minkowski plane (M 2 ,·) preserves Birkhoff orthogonality, then
(M 2 ,·) is a Radon plane.
Theorem: In a strictly convex Minkowski plane (M 2 ,·), let
G1,G2,G3 be three lines having a common point p. The plane
(M 2 ,·) is Euclidean if and only if there exists a line G4 through
p such that
ΨG3
◦ΨG2 ◦ΨG1 = ΨG4 .
Cycles in strictly
convex Minkowski
planes 15 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Bachmann’s concept on reflections and left-reflections
Cycles in strictly
convex Minkowski
planes 16 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Bachmann’s concept on reflections and left-reflections
Let R be the set of all left-reflections in a strictly convex
Minkowski plane (M 2 ,·), and let
R 2 := {Ψ2 ◦Ψ1 : Ψ1,Ψ2 ∈ R,} and
R 3 := {Ψ3 ◦Ψ2 ◦Ψ1 : Ψ1,Ψ2,Ψ3 ∈ R}.
Cycles in strictly
convex Minkowski
planes 16 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Bachmann’s concept on reflections and left-reflections
Let R be the set of all left-reflections in a strictly convex
Minkowski plane (M 2 ,·), and let
R 2 := {Ψ2 ◦Ψ1 : Ψ1,Ψ2 ∈ R,} and
R 3 := {Ψ3 ◦Ψ2 ◦Ψ1 : Ψ1,Ψ2,Ψ3 ∈ R}.
Theorem: In a strictly convex Minkowski plane (M 2 ,·) the set
R ∗ = R 3 ∪ R 2 is a group if and only if (M 2 ,·) is Euclidean.
Cycles in strictly
convex Minkowski
planes 16 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Bachmann’s concept on reflections and left-reflections
Let R be the set of all left-reflections in a strictly convex
Minkowski plane (M 2 ,·), and let
R 2 := {Ψ2 ◦Ψ1 : Ψ1,Ψ2 ∈ R,} and
R 3 := {Ψ3 ◦Ψ2 ◦Ψ1 : Ψ1,Ψ2,Ψ3 ∈ R}.
Theorem: In a strictly convex Minkowski plane (M 2 ,·) the set
R ∗ = R 3 ∪ R 2 is a group if and only if (M 2 ,·) is Euclidean.
Corollary: A strictly convex Minkowski plane is Euclidean if and
only if the left-reflections generate a proper closed subgroup of the
equiaffine group.
Cycles in strictly
convex Minkowski
planes 16 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Bachmann’s concept on reflections and left-reflections
Let R be the set of all left-reflections in a strictly convex
Minkowski plane (M 2 ,·), and let
R 2 := {Ψ2 ◦Ψ1 : Ψ1,Ψ2 ∈ R,} and
R 3 := {Ψ3 ◦Ψ2 ◦Ψ1 : Ψ1,Ψ2,Ψ3 ∈ R}.
Theorem: In a strictly convex Minkowski plane (M 2 ,·) the set
R ∗ = R 3 ∪ R 2 is a group if and only if (M 2 ,·) is Euclidean.
Corollary: A strictly convex Minkowski plane is Euclidean if and
only if the left-reflections generate a proper closed subgroup of the
equiaffine group.
Remark: Let (M 2 ,·) be strictly convex Minkowski plane and
S the set of reflections along all lines forming a pencil of parallel
lines. Then the S 3 ∪S 2 = S∪S 2 is a group isomorphic to the
group {x ↦−→ εx+b;b ∈ R,ε ∈ {−1,1}}.
Cycles in strictly
convex Minkowski
planes 16 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Further characterizations of the Euclidean norm
Cycles in strictly
convex Minkowski
planes 17 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Further characterizations of the Euclidean norm
Theorem: A strictly convex Minkowski plane (M 2 ,·) is
Euclidean if and only if the image of any circle with respect to any
left-reflection is also a circle.
Cycles in strictly
convex Minkowski
planes 17 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Further characterizations of the Euclidean norm
Theorem: A strictly convex Minkowski plane (M 2 ,·) is
Euclidean if and only if the image of any circle with respect to any
left-reflection is also a circle.
Theorem: A strictly convex Minkowski plane (M 2 ,·) is
Euclidean if and only if every left-reflection in (M 2 ,·) preserves
James orthogonality.
Cycles in strictly
convex Minkowski
planes 17 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Products of two left-reflections along intersecting lines
Remark: According to O. Veblen and J. W. Young:
Projective Geometry, Vol. 2, Blaisdell Publishing Co. Ginn and
Co., New York-Toronto-London, 1965, any element of the stabilizer
of a point p in the equiaffine group of the real affine plane is the
product of two affine reflections if it has the determinant 1, and of
one or three affine reflections if it has the determinant −1.
Cycles in strictly
convex Minkowski
planes 18 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Products of two left-reflections along intersecting lines
Theorem: Let (M 2 ,·) be a strictly convex Minkowski plane.
Then (M 2 ,·) is Euclidean if and only if for any two intersecting
lines G1 and G2 and an arbitrary line G ′ 1 passing through
through p such that
{p} = G1 ∩G2 there exists a line G ′ 2
Ψ G ′ 2 ◦Ψ G ′ 1 = ΨG2
◦ΨG1 ,
where ΨG is the left-reflections along the line G.
Cycles in strictly
convex Minkowski
planes 18 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Definition
Basic properties
Relations between
point reflections and
left-reflection
Characterizations of
special classes of
norms via
left-reflections
Bachmann’s concept
on reflections and
left-reflections
Further
characterizations of
the Euclidean norm
Products of two
left-reflections along
intersecting lines
Products of two left-reflections along intersecting lines
Theorem: In a strictly convex Minkowski plane (M 2 ,·) every
product of two left reflections is an isometry if and only if the
plane is Euclidean.
Cycles in strictly
convex Minkowski
planes 18 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Cycles in strictly convex Minkowski planes
19 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Definition
In a strictly convex Minkowski plane (M 2 ,·) let there be given
a point x and the pencil Gp of all lines passing through p = x.
The locus of points which are images of x with respect to the left
reflections along the lines of Gp is called the cycle of x with
respect to Gp, denoted by CY(p,x).
20 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Definition
In a strictly convex Minkowski plane (M 2 ,·) let there be given
a point x and the pencil Gp of all lines passing through p = x.
The locus of points which are images of x with respect to the left
reflections along the lines of Gp is called the cycle of x with
respect to Gp, denoted by CY(p,x).
20 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Properties of cycles in Minkowski planes
21 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Properties of cycles in Minkowski planes
Proposition: The cycle CY(p,x) contains a line segment if and
only if the unit circle C has a corner point.
21 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Properties of cycles in Minkowski planes
Proposition: The cycle CY(p,x) contains a line segment if and
only if the unit circle C has a corner point.
Proposition: If p = 0 and x = (1,0), then CY(0,(1,0)) =
{(t,+ 1−sin2t·b(t)+(b(t)) 2sin2 t; t ∈ [0,2π]}.
21 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Properties of cycles in Minkowski planes
Proposition: The cycle CY(p,x) contains a line segment if and
only if the unit circle C has a corner point.
Proposition: If p = 0 and x = (1,0), then CY(0,(1,0)) =
{(t,+ 1−sin2t·b(t)+(b(t)) 2sin2 t; t ∈ [0,2π]}.
Theorem: Any cycle CY(p,x) in a strictly convex Minkowski
plane (M 2 ,·) is a simple closed Jordan curve which is
starshaped with respect to x and 2p−x. Moreover, if CY(p,x)
contains no segment, then CY(p,x)\{x,2p−x} is a
differentiable curve.
21 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Non-convexity of cycles in Minkowski planes
22 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Non-convexity of cycles in Minkowski planes
The line y = s intersects the cycle CY(0,(1,0)) in the point
(cos2t0 − 1
2 b(t0)sin2t0,s) if and only if for the function b one has
b(t0) = s−sin2t0
sin2t0 . If t0 < t1 < ... < tn < π
2
are real numbers such
that for these real numbers the function cos2t− s−sin2t
2sin 2 t takes
different values and b(ti) = s−sin2ti
sin 2 ti
, then the points
(cos2ti − s−sin2ti
2sin2 ,s) are different points of CY(p,x). This shows
ti
that CY(p,x) is not convex since there are many differentiable
functions b satisfying these properties. The directions of the left
reflections with axes in Gp determine together the angles between
the directions and the tangent lines of the unit circle. This allows
to reconstruct the unit circle of a smooth Minkowski plane
belonging to the function b.
22 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of special norms via cycles
23 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of special norms via cycles
Theorem: Every two cycles CY(p,x) and CY(q,x) in a strictly
convex Minkowski plane (M 2 ,·), where p,q, and x are
collinear, have the unique point x in common if and only if the
plane (M 2 ,·) is smooth.
23 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of special norms via cycles
Theorem: Every two cycles CY(p,x) and CY(q,x) in a strictly
convex Minkowski plane (M 2 ,·), where p,q, and x are
collinear, have the unique point x in common if and only if the
plane (M 2 ,·) is smooth.
Theorem: Any cycle CY(p,x) in a strictly convex Minkowski
plane (M 2 ,·) is symmetric with respect to p if and only if
(M 2 ,·) is a Radon plane.
23 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of special norms via cycles
Theorem: Every two cycles CY(p,x) and CY(q,x) in a strictly
convex Minkowski plane (M 2 ,·), where p,q, and x are
collinear, have the unique point x in common if and only if the
plane (M 2 ,·) is smooth.
Theorem: Any cycle CY(p,x) in a strictly convex Minkowski
plane (M 2 ,·) is symmetric with respect to p if and only if
(M 2 ,·) is a Radon plane.
Theorem: Let x,y be two arbitrary different points in a strictly
convex Minkowski plane (M 2 ,·), and p be the midpoint of the
segment [x,y]. The cycles CY(p,x) and CY(p,y) coincide if and
only if (M 2 ,·) is a Radon plane.
23 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of the Euclidean norm via cycles
24 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of the Euclidean norm via cycles
Theorem: Let CY(p,x) be an arbitrary cycle in a strictly convex,
smooth Minkowski plane. Then, for any y ∈ CY(p,x), the relation
CY(p,x) ≡ CY(p,y)
holds if and only if the plane is Euclidean.
24 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of the Euclidean norm via cycles
Theorem: Let CY(p,x) be an arbitrary cycle in a strictly convex,
smooth Minkowski plane. Then, for any y ∈ CY(p,x), the relation
CY(p,x) ≡ CY(p,y)
holds if and only if the plane is Euclidean.
Theorem: A cycle CY(p,x) in a strictly convex Minkowski plane
M is a circle with center p if and only if the plane is Euclidean.
24 / 24

Motivation
Preliminares
Isometries in
Minkowski planes
and reflections in
the group of affine
transformations
Left-reflections in
Minkowski planes
Cycles in strictly
convex Minkowski
planes
Definition
Properties of cycles
in Minkowski planes
Non-convexity of
cycles in Minkowski
planes
Characterizations of
special norms via
cycles
Characterizations of
the Euclidean norm
via cycles
Characterizations of the Euclidean norm via cycles
Theorem: Let CY(p,x) be an arbitrary cycle in a strictly convex,
smooth Minkowski plane. Then, for any y ∈ CY(p,x), the relation
CY(p,x) ≡ CY(p,y)
holds if and only if the plane is Euclidean.
Theorem: A cycle CY(p,x) in a strictly convex Minkowski plane
M is a circle with center p if and only if the plane is Euclidean.
A lot of thanks for your attention!
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