# Questions tagged [convex-geometry]

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

762
questions

**3**

votes

**2**answers

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**

vote

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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**

vote

**0**answers

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**

votes

**1**answer

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**

vote

**0**answers

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**

vote

**1**answer

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.
...

**1**

vote

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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).
...