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A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
14
votes
2
answers
877
views
Lattice points and convex bodies
Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points c …
13
votes
2
answers
885
views
A problem on convex geometry
Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are antip …
13
votes
1
answer
438
views
Choquet theory and Hilbert's fourth problem
The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line segme …
13
votes
0
answers
494
views
Unit ball of smallest volume in a Hilbert geometry
In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body i …
10
votes
1
answer
505
views
Monotonicity of Loewner ellipsoid?
Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?
I have just finished proving a lem …
9
votes
1
answer
945
views
Reference request: Ehrhart's conjecture on the geometry of numbers
Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n …
9
votes
0
answers
373
views
An affine invariant of convex bodies
The following invariants of "pointed" convex bodies (i.e., pairs consisting of a convex body and a distinguished point in its interior) roughly measure how many of its linear images fit between the co …
8
votes
1
answer
402
views
From convex geometry to contact topology
Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine.
Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. …
8
votes
1
answer
366
views
Convex bodies with symmetric shadows
Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry.
This is a classic result …
7
votes
1
answer
710
views
A question on the Mahler conjecture
In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and
$$
K^* := \{y \in \mathbb{R}^n : \langle y, x …
6
votes
0
answers
159
views
Norms and distributions
Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := …
6
votes
0
answers
189
views
Variations on a problem of S. Mazur
In problem 76 of the Scottish Book Mazur asked
Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the lengt …
5
votes
1
answer
326
views
A question of compactness in the geometry of numbers
Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices tha …
3
votes
1
answer
196
views
Diameter of a convex body relative to its Legendre ellipsoid
Given a convex body in $\mathbb{R}^n$ that is symmetric with respect to the origin, let us measure its diameter with respect to the Euclidean metric determined by its own Legendre ellipsoid. How large …
3
votes
2
answers
154
views
Asymmetry of projections
A possible measure of asymmetry for a convex body $K \subset \mathbb{R}^n$ is the affine-invariant
quantity
$$
\alpha_n(K) := \frac{\textrm{vol}(K - K)}{2^n\textrm{vol}(K)} .
$$
Indeed, the Brunn-Mink …