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12
votes
2answers
250 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 ...
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 ...
1
vote
1answer
100 views

Relative interior and dense subsets

(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by ...
1
vote
1answer
115 views

Volume in tropical geometry as compared to volume in convex geometry

In tropical geometry, is there a notion of volume. Maybe one with some of the properties as found in classical convex geometry? If so, is there a good reference that elaborates on this question. I ...
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} = ...
5
votes
3answers
660 views

Set of Positive Definite matrices with determinant > 1 forms a convex set

While reading a paper An Arithmetic Proof of John’s Ellipsoid Theorem by Gruber and Schuster, I have a question on their proof. Consider an $n\times n$ real symmetric and positive definite matrix ...
6
votes
0answers
125 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 ...
1
vote
0answers
83 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 ...
2
votes
1answer
132 views

Algorithmic Version of John's Decomposition of Convex Body

While reading textbooks on convex geometry, I heard about Fritz John's theorem on convex body, which is known as John's Theorem or John's Decomposition. (I know that there are many variants, but this ...
1
vote
0answers
70 views

Is the size of $\varepsilon$-nets of the Euclidean ball exponential for large $\varepsilon$

Let $X$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_2)$. A finite set $N=N(\varepsilon)$ is a $\varepsilon$-net of $X$ if every point in $X$ is at most a distance $\varepsilon$ from a point from $N$. ...
0
votes
0answers
57 views

Uniform approximation of tangent cone

Let $\mathcal{C}$ be a convex and closed set in $\mathbb{R}^n$ containing $0$. The tangent cone of $\mathcal{C}$ at $0$ is given as, \begin{equation} T_{\mathcal{C}}(0)=Cl(\text{cone}(\mathcal{C})) ...
0
votes
0answers
96 views

Uniform convergence of difference quotients of a convex function

Let $f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function. At $x$ denote the subdifferential by $\partial f(x)$ which is compact and closed. Now, define the approximation of $f$ around a ...
11
votes
2answers
508 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 ...
0
votes
1answer
87 views

A question about rational convex cone

Edit: Fix a lattice $N = \mathbb{Z}^n$ and let $N_{\mathbb{R}} = N \otimes \mathbb{R}$. Let $C(S)$ be a strongly convex rational cone in $N_{\mathbb{R}}$ generated by a finite set $S \subset N$, with ...
4
votes
3answers
265 views

Can the intersection of the boundaries of compact and convex sets be a single element?

Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\bigcap_{j=1}^n H_j$ has non-empty interior and for each $i=1,2,\dots,n$ there exist at least one element $x \in H_i$ ...
7
votes
1answer
178 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?
1
vote
0answers
83 views

Extenstions of Urysohn's inequality

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| ...
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: ...
3
votes
2answers
193 views

Minkowski successive minima inequality for a lattice base?

Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors. The Minkowski successive minima inequality says ...
4
votes
0answers
110 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 $K_{1}+K_{2}$ ...
7
votes
2answers
355 views

Two cubes in unit cube

A cube of side one contains two cubes of sides a and b having non-overlapping interiors. How to prove the inequality $a+b \le 1$? The same question in higher dimensions. It was asked, but not answered ...
12
votes
1answer
340 views

(A question about)${}^3$ 3-dimensional convex bodies

Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...
1
vote
1answer
131 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 ...
6
votes
1answer
225 views

A question about a question about 3-dimensional convex bodies

For each positive integer n let E(n) denote n-dimensional Euclidean space and let the term "n-dimensional convex body" mean a compact convex subset of E(n) whose interior (with respect to E(n)) is ...
2
votes
1answer
222 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 $B_n$? ...
4
votes
6answers
401 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
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 ...
3
votes
1answer
103 views

Covering a convex body with its smaller homothetic copies

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
1
vote
1answer
148 views

Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
6
votes
1answer
234 views

Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example). One ...
6
votes
0answers
143 views

Minkowski's convex body theorem for ellipsoids

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector. Can this bound be improved ...
5
votes
0answers
131 views

Hilbert metric and cross-ratio of points on simplices

Background and motivation: Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (those for which $\sum ...
1
vote
0answers
148 views

A question on the theorem of Minkowski-Hlawka

The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. Then there is a ...
1
vote
4answers
2k views

How to find overlap between two convex hulls,along with the overlap area

I have two boundaries of two planar polygons , say,B1 and B2 of polygons P1 and P2(with m and n points in Boundaries B1 and B2). I want to find out if the Polygons overlap or not.If they overlap,then ...
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| \, ...
2
votes
2answers
194 views

Intersection points of straight line segment with Voronoi diagram

Hi, I need to find the x,y points of intersection of a vertical line with the edges of the Voronoi cells it goes through in a defined, rectangular plane region with a given Voronoi tesselation. Is ...
5
votes
1answer
276 views

A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{R}^n : \langle y, x ...
1
vote
2answers
192 views

Volume of a convex set

Say $K$ is a $n$-dimensional convex subset of $\mathbb R^n$ around the origin. Say we know $Vol_{n - 1}(\pi_{\theta^{\bot}}) K)$ where $\theta^\bot$ is the linear subspace of $\mathbb{R}^n$ orthogonal ...
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
1answer
276 views

Surface area of a convex set

In 2D perimeter(P) of a convex set around origin may be written as $P=1/2 \int m(\theta) d\theta$. Where $m(\theta)$ is the diameter of the set in the $\theta$ direction. This is related to ...
0
votes
1answer
183 views

Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
4
votes
1answer
172 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 a proving a ...
2
votes
1answer
94 views

Maximal cross sections of the Cartesian product of two planar domains

Let $K$ and $L$ be two two-dimensional convex bodies, and consider their Cartesian product $K\times L\subseteq \mathbb R^4$. Now let $U_\theta\subseteq \mathbb R^4$ be the two-dimensional subspace ...
4
votes
1answer
248 views

A question of compactness in the geometry of numbers

Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices ...
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 = ...
1
vote
2answers
146 views

What is the dual of an semidefinitely representable (SDR) cone?

The Question Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. Let $\mathcal K\subset V$ be a cone defined as $$ \mathcal K=\Big\{x\in ...
2
votes
2answers
207 views

the perimeter of a non-convex set

Let $\Omega$ be an open, bounded set in $\mathbb{R}^n$ with $C^1$ boundary. Is it true that the perimeter of the convex hull of $\Omega$ is smaller or equal the perimeter of $\Omega$ with equality if ...
4
votes
2answers
413 views

Expected distance of a random point to the convex hull of N other points

Let $X_1, \cdots, X_N$ be i.i.d. $d$-dimensional random vectors, the exact distribution of $X$ is not very important in my application (as long as it continuous) so pick the one that works best, but ...