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13
votes
2answers
933 views

Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection). I am wondering if each of the five ...
0
votes
0answers
23 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
0
votes
0answers
42 views

Integral representation formula for convex

For $u \in \mathbb{S}^{d-1} \subset \mathbb{R}^d$, it is easy to show that: \begin{equation} u=c_d \int_{\mathbb{S^{d-1}}} \xi \mathbb{1}_{\left\{x \cdot u >0 \right\}}(\xi) \ ...
23
votes
1answer
901 views

Which natural numbers are a square minus a sum of two squares?

Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$? This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...
1
vote
1answer
81 views

Linear map of Zonotopes [closed]

Consider a linear system with a map $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $y = A x$ with $n \geq m$ The input space $x$ is constrained by a zonotope set $\mathcal{X} \subseteq ...
5
votes
1answer
215 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 ...
0
votes
1answer
121 views

Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
6
votes
1answer
111 views

Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any, where we can know for sure that a convex body exists whose translative packing constant is strictly larger than its lattice ...
5
votes
2answers
176 views

Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...
14
votes
1answer
233 views

Does the boundary of a convex body contain a regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we ...
3
votes
3answers
271 views

Vertex-transitive polytopes in any dimension with any number of vertices?

Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...
6
votes
0answers
144 views

Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if ...
17
votes
10answers
7k 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 ...
0
votes
2answers
164 views

Determine the boundary points of a set of points [closed]

I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$? There are methods like convex hull, concave hull and ...
14
votes
4answers
411 views

The limit of edge-midpoint convex polyhedra

    Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$, replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$. Continuing this process, we obtain a ...
3
votes
2answers
148 views

A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors

The motivation for this question comes from this paper where we study lower bounds on the complexity of convex optimization algorithms (BTW, if you find a better title for my question, please let me ...
1
vote
0answers
34 views

About the $C^{1,1} $regularity of the boundary of a set

I am studying a paper that uses the following property : Consider $U$ and $V$ open, bounded and convex domains in $R^n$ with $U \supset \overline{V}$ and suppose that $min_{y \in V} |x - y| = \lambda ...
0
votes
1answer
83 views

About the regularity of the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial ...
2
votes
0answers
56 views

Regularity of the Minkowski functionnal of a convex

Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is: \begin{align*} \varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K ...
2
votes
0answers
71 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 ...
5
votes
2answers
338 views

Heaviest Convex Polygon

Suppose we have an arbitrary function $f : \mathbb{R}^2 \to \mathbb{R}$. For any subset $s \subseteq \mathbb{R}^2$, we can define $g_f(s)$ as the integral* of $f$ over the region $s$. Suppose ...
11
votes
3answers
635 views

Deciding membership in a convex hull

Problem: Given points $u,v_1,\dots,v_n\in\mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1,\dots,v_n.$ This can be done efficiently by linear programming (time polynomial in ...
4
votes
2answers
240 views

Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...
1
vote
2answers
1k views

Break Polyhedron into Tetrahedron

Given a polyhedron consists of a list of vertices (v), a list of edges (e), and a list of surfaces connecting those edges (...
0
votes
0answers
94 views

ellipsoids have spherical section

I want to prove that "For any $(2k-1)$-dimensional ellipsoid $E$ ,there is a $k$-flat $L$ passing through the center of $E$ such that $ E \cap L$ is a Euclidean ball. I see a proof for it in the book ...
1
vote
0answers
387 views

Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} \|x-y\|={\rm diam}(K). $$ where ${\rm diam}(K)$ denotes the ...
2
votes
1answer
105 views

Distance from constant width bodies

EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$. I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb ...
2
votes
1answer
192 views

Chebyshev centres of a bounded closed convex set in a strictly convex Banach space

Suppose $X$ is a strictly convex Banach space. Does there exist a bounded closed convex set $K$ in $X$ such that the set of all Chebyshev centers $C(K)$ of $K$ is a proper subset of $K$ with diameter ...
4
votes
0answers
116 views

Circumscribing simplex to convex body?

Q. Does every (perhaps smooth) compact convex body $K$ in $\mathbb{R}^d$ admit a circumscribing simplex, each facet of which touches (shares a point with) $K$? How about a circumscribing ...
3
votes
1answer
265 views

What is the shape of the convex $n$ -gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? ...
12
votes
1answer
637 views

Optimal 8-vertex isoperimetric polyhedron?

I know from Marcel Berger's Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry (p.531) that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
3
votes
0answers
53 views

Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...
3
votes
1answer
121 views

$\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$. Moreover, let $M \colon \mathbb{R}^n \to ...
3
votes
5answers
248 views

Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...
23
votes
3answers
663 views

Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
8
votes
1answer
231 views

Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
2
votes
0answers
70 views

Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations $C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$ The pair $(A,R)$ is referred to as a double description (DD) pair of the ...
4
votes
2answers
123 views

When does a cone contain its dual cone?

Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...
4
votes
6answers
432 views

A convex curve inside the unit circle

Has this theorem a specific name; and I need some references for general form. Suppose ${\gamma}$ is a convex polygon contained in the interior of the unit circle. then the length of $\gamma$ is not ...
1
vote
2answers
112 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
134 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
46 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
129 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
389 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
0answers
63 views

Decomposing a cone based on decompositions of its facets

Let $C$ be a cone in $\mathbb{R}^d$, and let $x_1, \dots, x_k$ be its extreme rays. Suppose that the $x_i$ satisfy: For all $i, j$, $\langle x_i, x_j \rangle \ge 0$, There is a partition $A \cup B ...
2
votes
1answer
75 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 ...
3
votes
1answer
227 views

Characterization of $l_p$ up to a linear isometry

There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach ...
0
votes
1answer
52 views

extreme points of the image of a nonlinear vector-valued function

Consider a continuous function $f : D \rightarrow \mathbb{R}^m$, where $D \subseteq \mathbb{R}^n$ is a compact convex set. I am in search of a result that helps me say something about the extreme ...
8
votes
1answer
183 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 ...
3
votes
0answers
69 views

Covering fat objects with fat objects

The family of rectangles has the cover property, i.e.: For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...