Questions tagged [convex-geometry]

A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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3
votes
2answers
259 views

Convex set with no interior contained in hyperplane?

Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane? It's fairly easy to see that this is true in $ℝ^n$, ...
1
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0answers
38 views

Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)

Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
3
votes
0answers
55 views

Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
3
votes
0answers
57 views

Ehrhart-Macdonald reciprocity with multiplicities

Let $P$ be a convex lattice polytope in $\mathbb{R}^n$. The function $L(t, P) = |\mathbb{Z}^n \cap t\cdot P|$ is a polynomial, and we have an equality $$L(-t, P) = (-1)^nL(t, P^{int}),$$ where $P^{int}...
14
votes
1answer
516 views

Acute triangles in “obtuse” polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute? I conjecture ...
3
votes
2answers
102 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-...
1
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0answers
37 views

ratio between the volume of a section of the cube and a section of the ball

Let $H\subset \mathbb{R}^n$ be a $k$-dimensional affine space and suppose $vol_k(H\cap [-\frac{1}{2},+\frac{1}{2}]^n)>0$ then can one upper bound the ratio $$\frac{vol_k(H\cap [-\frac{1}{2},+\frac{...
-2
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1answer
62 views

A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the ...
1
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0answers
45 views

Does it holds that the $L^{\infty}$ norm of the support function of a convex body is minimal on balls with the same volume? [closed]

I was wondering if the following inequality holds. Let $K$ be a convex body of $\mathbb{R}^n$ and let us denote by $h_K$ its support function, defined as, for $x\in\mathbb{R}^n$ $$ h_K(x)={\max}\{x\...
1
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1answer
112 views

On convex polygons contained in convex polygons

In what follows '$n$-gon' stands for '$n$-vertex polygonal region'. Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it. ...
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0answers
90 views

Integrality of polyhedra

Given two polyhedra in $H$ representation $P_1:Ax\leq b$ and $P_2:Bx\leq c$ which are integral are bounded when is their intersection also integral? Given two polyhedra in $H$ representation $P_1:Ax\...
2
votes
0answers
83 views

How to judge whether the following convex set contains a given point?

Let the set $\mathcal{S}=\left\{ \sum_{i=1}^n x_i\mathbf{h}_i:x_i\in[0,1] \text{ for all }i\right\}\subset\mathbb{R}^r$, i.e., a zonotope generated by $n$ column vector $\mathbf{h_1},\cdots,\mathbf{h}...
0
votes
0answers
64 views

Number of vertices in a polyhedron

Consider polytopes $$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$ $$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$ $$B[z_{1},z_{2},z]'\leq c$$ having vertex count $v_1,v_2$ and $v$ respectively. We ...
1
vote
1answer
107 views

Exactly counting number of vertices of a polyhedron

Suppose $Ax\leq b$ is a polyhedron, where the number of rows in $A$ is $r$, the vector $x$ lies in $\mathbb R^n$ and the rank of $A$ is $t$. Assume minimal number of hyperplane inequalities to define ...
3
votes
0answers
27 views

Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities: $$ \left<x,u_F\right> \geq -a_F$$ where $u_F\in \...
3
votes
0answers
60 views

Non-closed trajectories in convex billiards

This is a weak version of this problem, written down in Lviv Scottish Book. I start with necessary definitions. Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb ...
2
votes
1answer
152 views

Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary

I'm interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $\Omega\subset\mathbb{R}^n$. It seems that the following statement can be deduced from known results:...
1
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0answers
147 views

continuity of linear programming

I have the following conjecture: Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane ...
1
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0answers
25 views

How to express a polytope by a matrix inequalty? [duplicate]

Consider a convex V-polytope generated by the origin and $n$ points $\mathbf{h}_1,\cdots,\mathbf{h}_n$ in $\mathbb{R}^r$. A Theorem in the area of convex geometry shows that each V-polytope is a H-...
3
votes
0answers
97 views

A convex function is “usually” subdifferentiable

Let $X$ be a locally convex topological vector space, and let $f:X\to\mathbb R\cup\{\infty\}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(\mathbb R)$ is ...
9
votes
1answer
134 views

The set of boundary vectors of compact convex body has empty interior

Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$. Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
2
votes
1answer
136 views

Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles?

Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat ...
9
votes
0answers
115 views

A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
3
votes
1answer
100 views

Does convergence of convex sets in Hausdorff distance implies convergence of the complementary sets?

Definition: The Hausdorff distance associated with a distance $d$ on a space $E$ between two sets $A\subset E$ and $B \subset E$ is $d_H(A, B) = \max(\sup_{x\in B}\{d(x, A)\}, \sup_{y\in A}\{d(y, B)\})...
7
votes
1answer
114 views

The surface area measure in terms of support functions

$\def\RR{\mathbb{R}}$Let $K$ be a closed bounded convex body in $\RR^n$. The support function $h_K$ on $\RR^n$ is defined by $$h_K(v) = \max_{w \in K} \langle v,w \rangle.$$ Let $S^{n-1}$ be the unit ...
2
votes
1answer
71 views

Expected value of Tukey’s half-space depth for log-concave measures

Let ${\mathbb P}$ be a probability measure in ${\mathbb R}^n$. Let $x\in{\mathbb R}^n$ be an arbitrary point. Let ${\mathbb H}_x$ be the set of halfspaces of ${\mathbb R}^n$ containing $x$. Let \begin{...
4
votes
0answers
48 views

A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
3
votes
0answers
145 views

Sufficient condition for geodesic convexity/connectedness

Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
3
votes
1answer
67 views

Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists

The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as $$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
0
votes
0answers
19 views

Understanding non-convex subgradients and normal cones

I think I have a very good understanding of subgradients of convex functions and normal cones to convex sets. On the other hand, I have a lot of difficulties understanding them in the non-convex setup....
2
votes
3answers
236 views

Confining a polytope to one side of an affine hyperplane

Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem. This answer on math.stackexchange.com claims the ...
4
votes
0answers
107 views

“Baues poset” of shellings of simplicial polytope?

Let me start with some background I want to use as analogy. Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: ...
2
votes
1answer
56 views

Is the support function continuous on its effective domain?

Let $\sigma_D(x)=\sup \{ \left< x, y \right> : y\in D \}$ for a closed convex $D\subseteq \mathbb R^n$. Then $\sigma_D$ is convex and lower semicontinuous (it's the supremum of linear functions)....
0
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0answers
37 views

Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?

Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
6
votes
1answer
121 views

If a compact convex set meets the positive orthant does it meet it at a point with a normal in the positive orthant?

Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\...
2
votes
0answers
114 views

Helly-type theorem for infinite-dimensional spaces

Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. I'm a bit confused of authors use of a notions of Euclidean ...
2
votes
0answers
57 views

Weak convergence rates for integral operators

Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
0
votes
0answers
40 views

Dual cone operation is an antitone morphism of lattices

Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. The chapter on Dual cones made me thinkin of the fallowing ...
6
votes
0answers
61 views

A sufficient condition for being the boundary of one's convex hull?

Let $A\subset\mathbb R^n$ be such that: every non-zero linear functional is maximized by a unique point of $A$ every point of $A$ is a point where some linear functional achieves its maximum over $A$...
10
votes
2answers
281 views

Intersections and curvature in the plane

Let $D$ be a nonempty compact convex plane region whose boundary is a smooth curve whose radius of curvature is at most 1 everywhere. Can the boundary of $D$ intersect a circle of radius 1 in more ...
4
votes
0answers
48 views

circular disk in a convex domain

This question was asked on https://math.stackexchange.com/questions/4014421/circular-disk-in-a-convex-domain1 but got no reply Let $\partial \mathcal D$ be the smooth boundary of a convex domain $\...
4
votes
0answers
71 views

If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula ...
1
vote
0answers
95 views

Number of lattice points in a structural symmetric convex body

Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space \begin{equation} \...
1
vote
1answer
81 views

Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball [closed]

Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the ...
5
votes
1answer
124 views

What inequalities for convex sets are known since the work of Scott and Awyong?

In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
1
vote
1answer
63 views

Metric projection on closed convex sets in Busemann space

I am looking for a proof of the following statement: Let $X$ be a complete Busemann space. For any point $x\in X$ and any nonempty closed convex set $A\subseteq X$, there is a unique $a\in A$ such ...
8
votes
1answer
181 views

Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
23
votes
0answers
393 views

Can every 3-dimensional convex body be trapped by a tetrahedral cage?

Although the title question is fairly unambiguous, I give all relevant definitions: $\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-dimensional convex body if $C$ is convex, compact, and has non-...
-1
votes
1answer
103 views

A concave function as supremum of upper semi continuous is upper semi continuous

We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
1
vote
0answers
106 views

Angles between simple, closed geodesics on convex surface

It is known that there are at least three simple, closed geodesics on the surface of any smooth convex body $K$ in $\mathbb{R}^3$, the Lusternik-Schnirelmann Theorem (see links below for references). ...

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