Gruber P. Convex and Discrete Geometry

Noting that
2h 2 v − hhvv = 2π 2 a 2
K − cos π v
sin π v
v 6 sin 2 π v
I= 2a cos2 π v
v sin 2 π v
, h + vhv =
π
�v
0
g ′� h
cos2 ψ
a simple, yet lengthy calculation then shows that
MaaMvv − M 2 av
=−v 2 ha(2hvhav − hahvv)IJ + 2π 2ah2 a
v3 cos2 π JK −
v
=− 2π 2a cos π v
v5 sin3 π v
= 2π 2 a
v 5 sin 2 π v
= 2π 2 a
v 5 sin 2 π v
= 2π 2 a
v 5 sin 2 π v
π v
J
IJ+ 2π 2 a
v 5 sin 2 π v
�
K − cos π v
sin π I
v
�
J −
π
�v
0
�
× g ′� h
cos2 ψ
0
v
2a cos 2 π v
JK −
�
− 2π 2 a
v 5 sin 2 π v
�
K − cos π v
sin π I
v
33 Optimum Quantization 483
πa
v 2 sin 2 π v
π 2
v 4 sin 2 π v cos2 π v
v
2a cos 2 π v
g ′� h
cos2 ��
1 −
ψ
sin2 ψ
sin2 π �
dψ
cos
v
4 ψ
� 2 sin ψ
cos4 dψ > 0.
ψ
,
� sin 2 ψ
cos 4 ψ dψ(>0),
�
(ha + vhav)I − πha
v cos2 �2 π K
v
���
K − cos π v
sin π �
I
v
�
K − cos π v
sin π �2 I
v
�
K − cos π v
sin π �2 I
v
2a cos 2 π v
v sin 2 π v
Having proved that Maa and MaaMvv − M2 av are positive for a > 0,v ≥ 3, it follows
that the Hessian matrix of M is positive definite, which in turn implies (5), see the
convexity criterion 2.10 of Brunn and Hadamard.
Our next tool is the following simple consequence of Euler’s polytope formula.
See, e.g.
aiFejes Tóth, L. [329], p. 16:
(6) v1 +···+vn ≤ 6n.
Since, for fixed a, the function M(a,v) is convex in v by (5) and has a limit as
v →+∞(the moment of the circular disc with centre o and area a), we see the
following:
(7) For a fixed, M(a,v)is decreasing in v.