Gruber P. Convex and Discrete Geometry

D
10 Problems of Minkowski and Weyl and Some Dynamics 187
we see that
⎛
�
� �
� �ddx ⎜
(4) V (C) ≥ 1 − λ(x) = ⎝
=
�1
0
⎛
⎜
⎝
�
x∈D
λ(x)≤t
D
1
λ(x)
⎞
d (1 − t) d−1 ⎟
dx⎟
⎠ dt.
⎞
d (1 − t) d−1 ⎟
dt⎠
dx
Since z = λ(x) −1x ∈ bd D, itfollowsthatx∈λ(x)D. The condition that λ(x) ≤ t
is thus equivalent to the condition that x ∈ tD. Consequently,
⎛
�1
⎜
⎝
�
d (1 − t) d−1 ⎞
�1
⎟
dx⎟
⎠ dt = d (1 − t) d−1 V (tD) dt
0
x∈D
λ(x)≤t
0
�1
= V (D) d (1 − t) d−1 t d (d !)2
dt = V (D).
(2d) !
Substituting this back into (4) yields the desired upper bound for V (D). ⊓⊔
Remark. V (D) = 2d V (C) holds precisely when C is centrally symmetric and
V (D) = �2d� d V (C) holds precisely when C is a simplex, see [852]. In the proof of the
latter result the following characterization of simplices is needed: C is a simplex if
and only if C ∩ (C + x) is a non-negative homothetic image of C for each x ∈ Ed for
which C ∩ (C + x) �= ∅. This characterization of Rogers and Shephard [852] of simplices
slightly refines a classical characterization of simplices due to Choquet [208].
Compare Sect. 12.1.
For a stability version of the Rogers–Shephard inequality, see Böröczky Jr. [156].
A modern characterization of centrally symmetric convex bodies was given by
Montegano [751].
10 Problems of Minkowski and Weyl and Some Dynamics
The original versions of Minkowski’s problem [736, 739] and of Weyl’s problem
[1020] are as follows, where a closed convex surface is the boundary of a proper
convex body.
Problem 10.1. Given a positive function κ on S d−1 , is there a convex body C with
Gauss curvature κ (as a function of the exterior unit normal vector)?
Problem 10.2. Given a Riemannian metric on S d−1 , is there a closed convex surface
SinE d such that S d−1 with the given Riemannian metric is isometric to S if the
latter is endowed with its geodesic or intrinsic metric, i.e. distance in S is measured
along geodesic segments.
0