2
votes
0answers
36 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}_n$. Consider the real (positive) convex cone $\mathbb{R}^+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
1
vote
2answers
92 views

Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as $$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$ For ...
2
votes
1answer
116 views

intersection of the unit cube and a hyperplane containing the main diagonal

Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$, and consider the intersection of $A$ and the unit cube $\Delta_n$ ...
3
votes
0answers
40 views

Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then: $$ \max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y). $$ We can ...
4
votes
1answer
92 views

The number of facets of a polyhedron under linear transformation

Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets. Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Question1: Is there a fixed ...
5
votes
1answer
266 views

Linear transformation of a polyhedron

Is there a simple proof that shows: Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron. Minkowski sum of ...
2
votes
1answer
72 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
7
votes
1answer
174 views

Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
13
votes
0answers
339 views

Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
3
votes
0answers
99 views

Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...
0
votes
0answers
72 views

additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...
21
votes
4answers
563 views

Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...
6
votes
0answers
123 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 ...
2
votes
1answer
125 views

Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
6
votes
1answer
100 views

Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
1
vote
1answer
107 views

2-neighborhood of a simplex

Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I ...
7
votes
0answers
96 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
4
votes
0answers
162 views

Wasserstein distance, convex polytopes and extreme points

Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = ...
1
vote
0answers
82 views

Distance function from the origin to the boundary of a convex polytope

Let $K\subseteq \mathbb R^n$, $n\ge 3$ be a convex bounded polytope containing the origin in its interior. Consider the unit sphere with geodesic distance $(\mathcal S^{n-1},d)$ and define the ...
4
votes
1answer
192 views

n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
1
vote
3answers
132 views

How to determine if two rational cones intersect?

Let $$\displaystyle C_1=C(r_1,...,r_{n_1})=( x\in \mathbb{R^d}|x=\sum_{i=1}^{n_1}\lambda_i r_i, \lambda_i\in \mathbb{R_{>0}})$$ $$\displaystyle C_2=C(t_1,...,t_{n_2})=(x\in ...
5
votes
2answers
236 views

Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$? I found only one reference, to "Über einige ...
0
votes
2answers
233 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 ...
1
vote
1answer
202 views

non-convex Polytope definition

I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$, we can define a convex polytope in the following way: $$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, ...
3
votes
0answers
78 views

Covering points with a convex hull

Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose $\{x_1,\ldots,x_n\} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots,y_m \in C$ covers ...
2
votes
0answers
35 views

Conditions under which a set of points have a low weight representation under some basis

Let $S \subset R^d$ be a convex polytope in $d$ dimensional real space. Say that $S$ has weight $w$ if there exists some basis $x_1,\ldots,x_d$ such that for every point $v \in S$, we can write $v = ...
19
votes
4answers
572 views

Do random projections (approximately) preserve convexity?

The Johnson-Lindenstrauss lemma implies that any set of $k$ points in $\mathbb{R}^d$ can be randomly projected into $d' \approx \log(k)/\epsilon^2$ dimensions such that the distances between each pair ...
0
votes
0answers
82 views

centrally symmetric neighborly polytopes.

Let $P$ is centrally symmetric $k$ neighborly d-polytope. Let $ P^* $ is polar( also dual) of $ P $. Consider a polytope $ Q^* $ , $ Q^* \subset P^* $( not necessarily a face of $P^*$). By duality ...
2
votes
1answer
364 views

faces of a polytope

Let $a_1,a_2,..,a_n\in\mathbb{R}^m$, $n>m$ and the points are in general position meaning, no hyperplane contains more than $m$ points. Define a polyhedron $$P=\{x\in\mathbb{R}^m:a_1'x\leq0, ...
2
votes
2answers
524 views

Cone over the Join of two topological spaces

Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by ...
5
votes
1answer
323 views

Isometric embedding a convex cap to render its boundary planar

I would like to know if there is a polyhedral analog to this beautiful theorem of Hong: Theorem 11.0.1. Any smooth positive disk $(\bar{D},g)$ with a positive geodesic curvature along ...
0
votes
0answers
323 views

Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube

Let $C$ be a hypercube in $\mathbb{R}^D$ with edge length of $L$. Let $\mathcal{P}_1,\ldots,\mathcal{P}_K$ be $K$ convex polytopes that partition $C$. Let $S_k$ be the surface area of the polytope ...
1
vote
2answers
421 views

Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...
2
votes
2answers
467 views

Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center

Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan ...
2
votes
1answer
281 views

Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample

Definitions Let $\mathcal{C}$ be a convex polytope in $\mathbb{R}^{D}$ with $K$-facets $F_{1},\ldots,F_{K}$. I denote the normal vector of the $k^\mathrm{th}$ facet as ...
12
votes
3answers
627 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
3
votes
0answers
211 views

Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an element $C\in\mathcal C_d$ ...
35
votes
4answers
2k views

Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but noone had ever ...
8
votes
3answers
986 views

Listing lattice points in a simplex

Let $n \in \mathbf{Z}_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v_0,\dots,v_n \in \mathbf{Z}^n$, computes the set $\Delta ...
21
votes
4answers
1k views

Ellipse naturally associated with a polygon

My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
20
votes
2answers
1k views

The logic of convex sets

Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...
8
votes
1answer
191 views

Finding a point that lies in a majority of polytopes

Suppose I have $k$ $n$-dimensional polytopes $P_1,\ldots,P_k$, each explicitly specified as the intersection of a collection of hyperplanes. If there was a point $p \in \mathbb{R}^n$ that lay in the ...
0
votes
3answers
518 views

a different algebra/representation for convex sets

Hi, I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...
11
votes
5answers
722 views

A characterization of convexity

While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here. Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, ...
10
votes
0answers
647 views

Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup: Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
16
votes
3answers
1k views

Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference. Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...
3
votes
1answer
151 views

Edge-maximizing projective transformation on polytopes

Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let projn(e) denote the length of the projection of edge e onto vector n. Consider the set E of ...
5
votes
2answers
682 views

Maximal Ellipsoid

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
1
vote
1answer
374 views

Is the direction of the longest line of a polytope unique?

The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. The affine subspace is given by: $X \mbox{ u} = y$ where $u$ ...
14
votes
9answers
6k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...