# Tagged Questions

**5**

votes

**0**answers

74 views

### Geometry of the metric cone

Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...

**17**

votes

**2**answers

315 views

### An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that
$$
\frac{1}{\lambda} \left( \|x\|_1 ...

**13**

votes

**0**answers

380 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**5**

votes

**1**answer

140 views

### Shortest curve with given convex hull

Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...

**3**

votes

**1**answer

185 views

### Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...

**10**

votes

**1**answer

254 views

### Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$:
Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...

**6**

votes

**0**answers

131 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 ...

**12**

votes

**2**answers

229 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 ...

**6**

votes

**0**answers

119 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 ...

**4**

votes

**1**answer

170 views

### A construction related to scissors congruence

I was thinking about the following some time ago. My question is whether such things have been studied before.
Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...

**4**

votes

**0**answers

107 views

### Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...

**1**

vote

**1**answer

129 views

### A question concerning convex functions

Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for ...

**0**

votes

**2**answers

225 views

### Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...

**3**

votes

**1**answer

99 views

### What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...

**4**

votes

**2**answers

254 views

### How do you prove that every curve of constant width is convex?

Cross-posted from math.stackexchange:
Let C be a simple closed plane curve and let D be its interior. Recall that the width of C in a direction θ is the distance between two supporting lines for D ...

**4**

votes

**1**answer

172 views

### Extreme rays in the cone of (semi)metrics

How many extreme rays are there on the polytopal cone formed by all semimetrics on a set with $n$ elements?
Some background. Given a set $X$ with $n$ elements, the set of all semimetrics
$d:X \times ...

**8**

votes

**2**answers

200 views

### Can you see smoothness of the boundary of a convex body from its shadow?

Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?
The boundary $\partial K$ is a $C^1$-manifold if and only ...

**3**

votes

**1**answer

166 views

### Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...

**20**

votes

**1**answer

1k views

### A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter?
Background and motivation
The Borsuk conjecture (disproved in ...

**4**

votes

**1**answer

275 views

### Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...

**5**

votes

**0**answers

252 views

### Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...

**6**

votes

**2**answers

305 views

### Volume inequality between projections of a convex symmetric set in $\mathbb R^3$

Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K_F$ be the projection of $K$ in $\mathbb R^F$.
...

**10**

votes

**2**answers

1k views

### Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...

**13**

votes

**2**answers

522 views

### Do the elementary properties of mixed volume characterize it uniquely?

Background
Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a ...

**4**

votes

**1**answer

258 views

### Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity

Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...

**55**

votes

**0**answers

3k views

### Volumes of Sets of Constant Width in High Dimensions

Background
The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...

**8**

votes

**1**answer

466 views

### A variation on “Hearing the shape of a drum” for polytopes.

Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...

**0**

votes

**2**answers

2k views

### If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...

**17**

votes

**4**answers

1k views

### Minkowski sum of small connected sets

Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...