Questions tagged [convex-geometry]

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

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Extreme points of a two-dimensional convex body in terms of its surface area measure

Let $K \subset \mathbb{R}^2$ be a nonempty compact convex set. For any $t \in S^1$, define the unit vector $u_t = (\cos t, \sin t)$ making an angle of $t$, and let $l_K(t)$ be the tangent line of $K$ ...
2 votes
1 answer
306 views

Tangent cone of a closed convex cone

Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by) $$ T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
4 votes
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good textbooks which deal with convex geometry, especially convex polyhedral cones and convex polytopes

I study toric varieties. In toric geometry, we use convex geometry, especially convex polyhedral cones and convex polytopes. Are there good textbooks which deal with convex geometry, especially convex ...
5 votes
1 answer
216 views

Is the interior of the tensor product of two convex cones equal to the tensor product of their respective interiors?

I am sorry that the following question is elementary. I have not received an answer from my post at Math Stack Exchange. In the following question, all cones are convex and contain the origin. Let $C \...
4 votes
2 answers
217 views

An upper bound of gradient norm for convex functions near minimizer

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...
2 votes
2 answers
2k views

Finding points inside innermost convex hull [closed]

Given a set of points $S$ on the Euclidean plane, Onion Peeling determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in $S$,...
0 votes
0 answers
22 views

Number of cliques of intersection graphs in hyperconvex metric spaces

(cross-posted from Math.SE) I would like to find out if it is known whether intersection graphs of closed unit balls in hyperconvex metric spaces have polynomially many cliques. A metric space is ...
1 vote
0 answers
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Expectation of dual norm induced by probability measure

Let $\mu$ be a probability measure supported on the unit sphere $\mathbb{S}^{d-1}$. Assuming that $\mu$ is even and not supported on any great subsphere, its cosine transform $w \in \mathbb{R}^d \...
1 vote
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Partitioning convex regions, maximizing the average perimeter of pieces

We continue from Cutting convex regions into equal diameter and equal least width pieces - 2 Question: If a planar convex region C is to be cut into n convex pieces such that the average of the ...
10 votes
1 answer
316 views

covering convex sets by round balls

Let $X=\{x_1,...,x_k\}\subset E^n$ be a finite subset in the Euclidean $n$-space, $r>0$ and $B(x_i,r)$ are open balls of radius $r$ centered at the points $x_i\in X, i=1,...,k$. Suppose that $$ \...
0 votes
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Universal covering problem in dimension three

In dimension two one can quickly prove that every shape of diameter $1$ can be covered with a regular hexagon of width $1$. Thus, one may ask what is the minimal area of such a "universal ...
4 votes
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Quantifying error in the reconstruction of convex polytopes from moments

The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
4 votes
1 answer
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Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
1 vote
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Fitting a convex polytope with 𝑛 facets between two nested spheres

This is related to a research problem that is interested in approximation of spheres by convex polytopes. Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
1 vote
0 answers
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Inside-out dissections of polygons - a generalization

Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...
4 votes
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Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...
5 votes
1 answer
203 views

The bounded complex of a polyhedral decomposition

Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties: The union ...
7 votes
1 answer
525 views

Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

I have difficulty even in finding a Russian version of the next paper: "Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex function and some properties of ...
0 votes
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On convexity of special fractals in the plane

Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$. For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...
2 votes
0 answers
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Removing convex sets that are unions of other convex sets from a large combinatorial enumeration

An addition chain for $n$ is a sequence of integers $1=a_0<a_1<...<a_r=n$ with $a_i=a_j+a_k,i>j\ge k\ge 0$. We say $r$ is the addition chain length. We define the length of the smallest ...
1 vote
0 answers
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?

Suppose that $P$ is a polyhedron represented by $$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$ and $P$ contains interior points. Moreover, the ...
0 votes
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70 views

The closest ellipse and circle to a given triangle - 2

We add a little more to The closest ellipse to a given triangle. The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are. In an earlier post - ...
11 votes
2 answers
521 views

Convex hull of the Stiefel manifold with non-negativity constraints

Consider the Stiefel manifold $$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$ where $I_k$ is the $k$-dimensional identity matrix. It is well known that $$\mathrm{conv} \left( ...
5 votes
2 answers
130 views

Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?

Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
14 votes
4 answers
445 views

Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$. What is known about $V_n$? Is there a ...
0 votes
1 answer
88 views

Maximum number of vectors with bounds on inner products (follow up question)

This is a follow-up question from my previous question. Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
3 votes
2 answers
202 views

Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles?

Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat ...
1 vote
1 answer
141 views

Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?

Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function. Let $X$ be log-concave ...
2 votes
0 answers
200 views

Toric decomposition of multipartitions

Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$. Let's call $\lambda$ ...
0 votes
0 answers
100 views

Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where $$ E(r) := E \cap B_2^d(r)...
7 votes
2 answers
268 views

Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
8 votes
1 answer
207 views

Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...
1 vote
1 answer
88 views

abstract description of the topology on a real vector space defined by the algebraically open sets

Let $V$ be a real vector space. Given a subset $A \subseteq V$, say that a point $x \in A$ lies in the algebraic interior of $A$ if every affine line $\ell$ that passes through $x$ has the property ...
2 votes
1 answer
82 views

On equipartitions of surfaces of 3D convex regions

Let S be the surface of a 3D convex region (a 'convex surface'). Let S' be a subset of S. We shall refer to S' as geodetically convex in S if the following condition holds: If A and B are two points ...
4 votes
0 answers
127 views

Can a convex frame hold all circles of radius $1/n$ immobile?

Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile. By "immobile", I mean no circle can move without overlapping other circles or the frame, ...
18 votes
3 answers
2k 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
0 answers
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Anti-concentration of the $\ell_2$ norm of log-concave measures

This question is regarding a special case of this question, for which it is plausible the details are known. The Carbery-Wright inequality is an "anti-concentration inequality" that states ...
0 votes
1 answer
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On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry) Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
4 votes
1 answer
101 views

Mean width of a simplex as one edge becomes longer

Consider an edge $e$ of a simplex $C$. If we increase the length of $e$ while keeping the length of other edges constant, the width of $C$ increases in some directions and decreases in other ...
2 votes
1 answer
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'Constrained morphing' of planar convex regions

Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints. Qn: If $C_1$ and $...
2 votes
1 answer
165 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 R^n\ ...
4 votes
0 answers
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On moments of inertia of planar and 3D convex bodies

The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one ...
1 vote
0 answers
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Possible extensions of the perpendicular axes theorem for moment of inertia

This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia. The perpendicular axis theorem states that the moment ...
0 votes
0 answers
107 views

Question about symmetric bilinear form and convex geometry

Consider a Finsler norm $\varphi$ in $R^4$, and a positive definite symmetric bilinear form $S$, there also exists a vector-valued symmetric linear form $Q$, such that if $S(v, w) = 0$, then $Q(v, w)$ ...
5 votes
1 answer
184 views

Improving regularity of the boundary of a convex set in Riemannian manifolds

Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$...
12 votes
1 answer
318 views

What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible ...
8 votes
1 answer
1k views

Differentiability of distance to a closed convex set [closed]

Let $( \mathbb{R}^d, \| \mathbf{x}\|_2 )$ be a Euclidean Space. For any nonempty closed convex set $A\subseteq \mathbb{R}^d$, we define \begin{align} d(\mathbf{x}, A) = \inf \{ \| \mathbf{x} - \mathbf{...
5 votes
1 answer
273 views

Existence of geodesic convex functions

By a result of Shing-Tung Yau [1974, Mathematische Annalen 207: 269-270], there are no non-trivial continuous geodesic convex functions on complete manifolds with finite volume. What happened if we ...
2 votes
0 answers
199 views

Geometric inequality related with convexity of the boundary

I'm new to Mathoverflow, so hopefully my question is well-posed. My problem states as follows: Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...
0 votes
0 answers
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Number of tiles inside a region of a hyperbolic tiling

Let $\mathbb{D}$ be the Poincare disc model of hyperbolic geometry with $\{p,q\}$ tiling on it. In this, paper authors calculated the number of tiles on any circular region centered at a vertex or ...

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