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55
votes
0answers
3k views

Volumes of Sets of Constant Width in High Dimensions

Background The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...
36
votes
4answers
2k views

Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but noone had ever ...
30
votes
0answers
901 views

Two-convexity ⇒ Lefschetz?

Assume that $\Omega$ is an open simply connected set in $\mathbb R^n$ (two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$. Is it true that any component of ...
22
votes
3answers
2k views

Covering a circle with red and blue arcs

We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter ...
21
votes
1answer
461 views

The maximal “nearly convex” function

The following problem is only tangently related to my present work, and I do not have any applications. However, I am curious to know the solution -- or even to see a lack thereof, indicating that the ...
20
votes
1answer
1k views

A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter? Background and motivation The Borsuk conjecture (disproved in ...
19
votes
5answers
10k views

Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...
18
votes
2answers
363 views

Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...
17
votes
1answer
725 views

If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?

This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...
17
votes
4answers
1k views

Minkowski sum of small connected sets

Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...
17
votes
2answers
371 views

An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that $$ \frac{1}{\lambda} \left( \|x\|_1 ...
15
votes
12answers
3k views

Generalizations of the Birkhoff-von Neumann Theorem

The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices. The question is to point out different ...
14
votes
0answers
420 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
13
votes
2answers
455 views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
13
votes
2answers
260 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 ...
13
votes
2answers
548 views

Do the elementary properties of mixed volume characterize it uniquely?

Background Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a ...
13
votes
2answers
446 views

Extreme points of unit ball in tensor product of spaces

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively. Let $e(B_1), e(B_2)$ be corresponding extreme points sets. Consider the unit ball $B$ in tensor product ...
12
votes
0answers
129 views

Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...
12
votes
0answers
1k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers ...
11
votes
5answers
727 views

A characterization of convexity

While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here. Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, ...
11
votes
2answers
406 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 ...
11
votes
2answers
795 views

convex hull of k random points

suppose we have $k$ points placed uniformly at random in the unit cube in $\mathbb{R}^n$. what is the probability that their convex hull has all of the $k$ points as extreme points? [if it would be ...
11
votes
2answers
2k views

Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
11
votes
1answer
405 views

Determination of a symmetric convex region by parallel sections

This question is partly inspired by a problem in Stewart's Calculus: "Find the area of the region enclosed by $y=x^2$ and $x=y^2$." Suppose $f\colon [0,1]\to [0,1]$ is a convex increasing function ...
11
votes
1answer
346 views

Convex cones and self-duality

Consider the Euclidian space $E_n={\mathbb R}^n$, with standard scalar product $$x\cdot y=x_1y_1+\cdots+x_ny_n.$$ A closed convex cone $\Gamma\subset E_n$ defines an order by $y\ge x$ iff ...
11
votes
0answers
169 views

How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a ...
11
votes
1answer
419 views

Status of the compact AR problem?

The so-called "compact AR Problem" reads: Is every compact convex set in a metrizable topological vector space an absolute retract? It is open according to the chapter by T. Banakh, R. Cauty ...
10
votes
2answers
327 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
10
votes
1answer
297 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
10
votes
1answer
369 views

convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t ...
9
votes
3answers
1k views

Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?
9
votes
2answers
214 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 ...
9
votes
3answers
600 views

A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. ...
8
votes
1answer
753 views

Extreme points of a set of probability measures

Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int ...
8
votes
2answers
106 views

Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ : $$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$ The discriminant of this polynomial in $r$ is the following expression (obtained using ...
8
votes
3answers
439 views

surfaces of constant centro-affine curvature

It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by this question, I'd like to ask whether a similar rigidity holds for ...
8
votes
1answer
942 views

Banach-Mazur distance between $\ell^p$-norms

Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then $$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$ is an operator norm ...
8
votes
1answer
553 views

volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product. Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
8
votes
1answer
414 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
8
votes
2answers
844 views

Does a notion of convex graph make sense?

Let $X=(V,E)$ be a finite connected graph. I would be interested in some notion of convexity. General question: Is there a notion of convexity for finite connected graphs? How does it look like? ...
8
votes
1answer
469 views

A variation on “Hearing the shape of a drum” for polytopes.

Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...
8
votes
1answer
539 views

Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...
8
votes
0answers
294 views

What is the “positive part” of the unit ball in $M_n(R)$ ?

In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm $$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$ where $\|x\|$ is the Euclidian norm. The closed unit ball $B$ is the set of contractions (in the ...
7
votes
9answers
3k views

Ways to prove an inequality

It seems that there are three basic ways to prove an inequality eg $x>0$. Show that x is a sum of squares. Use an entropy argument. (Entropy always increases) Convexity. Are there other means? ...
7
votes
3answers
268 views

Characterising semi-definite positiveness on vectors with non-negative entries

My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
7
votes
2answers
266 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 ...
7
votes
0answers
218 views

Are plactic classes convex under the right weak Bruhat order?

For those who are unfamiliar with the terminology, I'll explain a little. The symmetric group $S_n$, as a type A Coxeter group, has generators $\{s_1,\ldots,s_{n-1}\}$ with relations (1) $s_i^2$ for ...
6
votes
4answers
381 views

Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks. If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...
6
votes
2answers
311 views

Volume inequality between projections of a convex symmetric set in $\mathbb R^3$

Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K_F$ be the projection of $K$ in $\mathbb R^F$. ...
6
votes
1answer
3k views

Eigenvalues of product of two symmetric matrices

This is mostly a reference request, as this must be well known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or ...