Gruber P. Convex and Discrete Geometry

150 Convex Bodies
Integration implies that
n=1
x ′ (s) 2 + y ′ (s) 2 = 1for0≤ s ≤ 2π.
�2π
0
x ′ (t) 2 dt +
�
0
2π
y ′ (t) 2 dt = 2π.
Consider the Fourier series for x(·) and y(·),
x(s) = a0
2 +
∞�
(an cos ns + bn sin ns), y(s) = c0
2 +
∞�
(cn cos ns + dn sin ns).
Then
x ′ ∞�
(s) = (nbn cos ns − nan sin ns), y ′ ∞�
(s) = (ndn cos ns − ncn sin ns).
n=1
The formula of Leibniz to calculate the area of a planar set bounded by a closed
Jordan curve of class C 1 and a version of Parseval’s theorem then yield the following.
and
L 2 = 4π 2 ⎛
�
= 2π ⎝
Thus
L 2 − 4π A = 2π 2
A =
0
= 2π 2
2π
�2π
0
x(t)y ′ (t)dt = π
x ′ (t) 2 dt +
�
0
2π
y ′ (t) 2 dt
n=1
n=1
∞�
n(andn − bncn)
n=1
⎞
⎠ = 2π 2
∞�
n 2 (a 2 n + b2 n + c2 n + d2 n ).
n=1
∞� � 2 2
n (an + b 2 n + c2 n + d2 n ) − 2n(andn − bncn) �
n=1
∞� �
(nan − dn) 2 + (nbn + cn) 2 + (n 2 − 1)(c 2 n + d2 n )� ≥ 0,
n=1
where equality holds if and only if
In this case
cn = dn = an = bn = 0forn ≥ 2, a1 = d1, b1 =−c1.
x(s) = a0
2 + a1 cos s + b1 sin s,
y(s) = c0
2 − b1 cos s + a1 sin s.
This is the parametrization of a circle – note that
�
x(s) − a0
2
� 2
�
+ y(s) − c0
2
� 2
= a 2 1 + b2 1 . ⊓⊔