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### 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 ...

**51**

votes

**2**answers

848 views

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

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ 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 ...

**36**

votes

**4**answers

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### 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 ...

**32**

votes

**0**answers

1k 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 ...

**29**

votes

**6**answers

17k 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 ...

**22**

votes

**13**answers

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### 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 ...

**22**

votes

**3**answers

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

**1**answer

502 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 ...

**21**

votes

**1**answer

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 1993) ...

**19**

votes

**2**answers

397 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 ...

**18**

votes

**1**answer

846 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, ...

**18**

votes

**4**answers

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

**2**answers

754 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

**3**answers

713 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 ...

**15**

votes

**2**answers

712 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 ...

**15**

votes

**1**answer

610 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...

**15**

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**0**answers

217 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, ...

**14**

votes

**1**answer

8k 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 $BA$...

**14**

votes

**1**answer

2k 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 $...

**14**

votes

**0**answers

272 views

### A functional inequality about log-concave functions

Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...

**14**

votes

**0**answers

1k 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

**2**answers

454 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

**2**answers

3k 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 $X$. ...

**13**

votes

**2**answers

541 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 $...

**13**

votes

**1**answer

608 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 $y-x\in\...

**12**

votes

**5**answers

781 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 $P$, $P\...

**12**

votes

**1**answer

370 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 - ...

**12**

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**0**answers

208 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

**2**answers

984 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

**1**answer

410 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

**1**answer

415 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 \...

**11**

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**2**answers

943 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 \...

**11**

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**1**answer

299 views

### Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...

**11**

votes

**1**answer

429 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

**2**answers

280 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 ...

**10**

votes

**2**answers

395 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

**3**answers

620 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. Specifically,...

**10**

votes

**2**answers

651 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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

**9**

votes

**3**answers

2k 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

**1**answer

485 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 ...

**9**

votes

**2**answers

1k 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?
...

**9**

votes

**0**answers

153 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

**8**

votes

**2**answers

162 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

**1**answer

1k 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 x\...

**8**

votes

**3**answers

475 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 centro-...

**8**

votes

**1**answer

371 views

### Generalization of Popoviciu's inequality

Popoviciu's inequality states that for convex $f$ and numbers $x_1,x_2,x_3$, we have
$f(x_1)+f(x_2) + f(x_3) + 3\cdot f(\frac{x_1+x_2+x_3}3) \geq 2\cdot f(\frac{x_1+x_2}2)+2\cdot f(\frac{x_1+x_3}2)+2\...

**8**

votes

**3**answers

271 views

### Shape whose translated and scaled copies are closed under intersection

The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation.
What ...

**8**

votes

**1**answer

1k 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

**2**answers

221 views

### Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...

**8**

votes

**1**answer

510 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 ...