The convex-geometry tag has no wiki summary.

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### Convex hull of the union of two parameterized curves in $\mathbb{R}^3$

My goal is to find a way to calculate the convex hull of the union of some parameterized curves.
For instance, I had to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup ...

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25 views

### Unimodular triangulation and affine toric variety

Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$.
Here, pointed means that all $r_i$ lie strictly on one side of ...

**3**

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**0**answers

41 views

### max volume of inscribed simplex in a ball

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\epsilon)$ be the spherical cap with height $\epsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in ...

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vote

**1**answer

159 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 ...

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**0**answers

41 views

### Lower bound for Quermassintegrals

Let $K$ be a convex body (a compact, convex subset of $R^n$ with empty interior), denote $W_0(K),W_1(K),\cdots,W_n(K)$ the Quermassintegrals of $K$. Note that $W_0(K)=V(K)$ the volume of $K$, and ...

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votes

**2**answers

206 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 ...

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**1**answer

69 views

### two-dimensional sections of polyhedral cones

Given a polyhedral cone, its intersection with any two-dimensional plane is either a polygon or a region enclosed by a polygonal curve. Is it a characterization of polyhedral cones? Does there exists ...

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**0**answers

69 views

### Positive curvature of the boundary away form a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...

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**0**answers

108 views

### Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...

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**0**answers

20 views

### Looking for a homogeneous function with some properties

I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...

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votes

**2**answers

774 views

### Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...

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**1**answer

239 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 ...

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votes

**2**answers

279 views

### Lipschitz parametrization of a symmetric convex curve

Assume that $\gamma$ is a convex curve which is symmetric with respect to two orthogonal lines (the ellipse is such a curve).
I want to know if there exists a $(l,L)$-bi-Lipschitz mapping of the ...

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votes

**2**answers

969 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 ...

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**0**answers

27 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}
...

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**0**answers

45 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

**1**answer

946 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 ...

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vote

**1**answer

89 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 ...

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votes

**1**answer

228 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 ...

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**1**answer

119 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 ...

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votes

**2**answers

187 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 ...

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**1**answer

293 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 ...

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votes

**3**answers

278 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 ...

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**10**answers

8k 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 ...

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**2**answers

198 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 ...

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votes

**4**answers

426 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 ...

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votes

**2**answers

156 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 ...

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44 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 ...

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**1**answer

94 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 ...

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**0**answers

60 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 ...

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73 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 ...

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**2**answers

339 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 ...

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votes

**3**answers

650 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 ...

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votes

**2**answers

260 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 ...

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**2**answers

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 (...

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108 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 ...

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**0**answers

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

**1**answer

106 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 ...

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votes

**1**answer

201 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 ...

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118 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 ...

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votes

**1**answer

268 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 $B_n$? ...

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**1**answer

650 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 ...

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votes

**0**answers

54 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 ...

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**1**answer

134 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 ...

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**5**answers

268 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 ...

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**3**answers

675 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 $\mathbb{R}^3$?

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**1**answer

236 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?

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### 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 ...

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votes

**2**answers

131 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 ...

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votes

**6**answers

475 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 ...