A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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1answer
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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
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2answers
277 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 non-...
3
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1answer
286 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 $B_n$? $$B_n=\sum_{1\le{i}\...
6
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6answers
616 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 ...
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3answers
239 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 \mathbb{R^d}|x=\sum_{i=...
3
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2answers
290 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 ...
3
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1answer
137 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 ...
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1answer
157 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
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1answer
290 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 ...
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0answers
190 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 ...
6
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1answer
194 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 ...
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0answers
261 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 ...
4
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4answers
13k 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, ...
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2answers
255 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 ...
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2answers
271 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 $E(v)=...
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1answer
627 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
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2answers
422 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
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1answer
419 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 \...
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2answers
286 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 ...
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0answers
117 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
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1answer
554 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 Cauchy-...
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1answer
199 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 ...
6
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1answer
277 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 ...
2
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1answer
106 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
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1answer
262 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 ...
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0answers
44 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 = \...
2
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2answers
187 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 V\ \...
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2answers
460 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
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2answers
507 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 ...
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0answers
220 views

Does there always exist a self dual polytope that contains a given polytope contained in its dual?

Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...
2
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1answer
264 views

another diameter-perimeter-area inequality

Recently I learnt that $$ \inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no non-...
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2answers
319 views

a diameter-perimeter-area inequality for convex figures

Is the following inequality known? I believe it's true, but I could find no reference. For any convex body $C$ in the plane we have $$\left(4-\frac{8}{\pi}\right)area(C)\leq > diam(C)(per(...
22
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3answers
2k views

Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...
2
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1answer
320 views

A quantitative version of Straszewicz's theorem?

Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an exposed point of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are ...
4
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0answers
177 views

Convex bodies with symmetric shadows.

Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry. This is a classic result ...
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0answers
345 views

Volume-like property to upper bound lattice points in a convex body

The following question arises in passing in a joint paper that I am working on. Let $K$ be a centrally symmetric convex body in an $n$-dimensional real vector space $V$ which contains a lattice $L$. ...
4
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1answer
467 views

An elementary probability question

Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite. Consider $n+1$ samples $X_0, \ldots, X_n ...
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0answers
221 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
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2answers
488 views

A problem on convex geometry

Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are ...
10
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2answers
281 views

Can you see smoothness of the boundary of a convex body from its shadow?

Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true? The boundary $\partial K$ is a $C^1$-manifold if and only ...
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2answers
548 views

Nonzero convex combinations of convex hull vertices to yield an inner point

Two questions: 1) (ALREADY ANSWERED) This is likely to be a very basic question for you folks. Carathéodory's theorem gives us an upper bound for the minimum number of convex hull vertices that can ...
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0answers
128 views

Almost Isodiametric Sets

Hi, The isodiametric inequality tells us that, of all sets of diameter $r$, the one with the largest Lebesgue measure is the ball of radius $r/2$ - and this holds regardless of norm. Let $\tau(r)$ be ...
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1answer
652 views

Is minimum of convex envelope the same as minimum of the original function?

Hello everyone my question is: $Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the ...
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2answers
275 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
21
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4answers
763 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 ...
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1answer
346 views

Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere $S$ in $\mathbb{R}^3$, sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$, with $\alpha < 1$ the fraction of ...
5
votes
5answers
695 views

Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid

A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...
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0answers
207 views

Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only one orientation of stable equilibrium and one orientation of unstable ...
3
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2answers
491 views

The (Sigma) Algebra of Convex Sets

This is a question-by-proxy for a colleague from computer science. I'm sure many people here are already aware that convex decomposition forms an important sub-field of both computational geometry and ...
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0answers
182 views

A linear program related question

Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice. Let $\alpha^k \in (\alpha_1^k, \alpha_2^...