The convex-geometry tag has no wiki summary.

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**1**answer

353 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in ...

**3**

votes

**0**answers

83 views

### If $\angle 0xy\leq \pi/2$ for every $x\in \partial K$, $y\in K$, then $K$ is a ball

Let $K$ be a bounded closed convex set in $\mathbb{R}^d$, $0$ lies in interior of $K$. Assume that for every boundary point $x$ of $K$ the plane through $x$ perpendicular to $x$ is a support plane of ...

**7**

votes

**1**answer

124 views

### Upper bound for the number of integral points in a convex set

Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer
coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?

**4**

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**0**answers

73 views

### Approximation of convex body by polytopes

In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$,
$$\rho_H(U,\mathcal{P}_n)\leq c(U) ...

**4**

votes

**1**answer

108 views

### 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 ...

**3**

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**0**answers

45 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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**0**answers

42 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 ...

**6**

votes

**1**answer

75 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 ...

**3**

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115 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 ...

**0**

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**0**answers

22 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 ...

**0**

votes

**0**answers

29 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

46 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

952 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

**1**answer

112 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

**2**answers

190 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 ...

**16**

votes

**1**answer

302 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 ...

**6**

votes

**1**answer

240 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 ...

**0**

votes

**2**answers

235 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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vote

**0**answers

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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votes

**4**answers

434 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 ...

**0**

votes

**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 ...

**3**

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**2**answers

158 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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**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 ...

**4**

votes

**2**answers

263 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 ...

**6**

votes

**1**answer

122 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 ...

**0**

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**0**answers

112 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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**1**answer

107 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 ...

**4**

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**0**answers

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 ...

**2**

votes

**1**answer

205 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 ...

**3**

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**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 ...

**3**

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**1**answer

140 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

275 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 ...

**3**

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**0**answers

94 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 ...

**5**

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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**0**answers

76 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

**2**answers

134 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

**1**answer

146 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**

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**0**answers

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 ...

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votes

**1**answer

138 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

**1**answer

457 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

**1**answer

78 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

**1**answer

234 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 ...

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**1**answer

186 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

**0**answers

71 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) ...

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**0**answers

346 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?

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131 views

### Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...

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**0**answers

124 views

### A question on Hilbert geometries as metric-measure spaces

Recall that a Hilbert geometry is the interior of a convex body $K \subset \mathbb{R}^n$ provided with the metric
$$
d(x,y) = \frac{1}{2} \ln\left(\frac{|x-b|}{|y-b|}\frac{|y-a|}{|x-a|}\right) ,
$$
...

**1**

vote

**1**answer

129 views

### Vertices of Curves and Eigenvectors of Hessian

This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to ...

**0**

votes

**1**answer

57 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 ...

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**2**answers

211 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 ...