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2
votes
0answers
226 views

Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?

Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with weak-$*$ topology (weak topology induced by the continuous functions). Consider a ...
0
votes
1answer
232 views

Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
3
votes
2answers
470 views

A lower bound of a particular convex function

Hello, I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where ...
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 ...
0
votes
0answers
127 views

Convexity of a cone in CxC

Hi. I have trouble deciding if the set of couples $ \left( \xi, \zeta \right) \in \mathbb{C}^2 $ with $ Re \left( \xi \text{ } \overline{\zeta} \right) > 0 $ is convex. It is a (real) cone, but is ...
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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
2
votes
2answers
428 views

Existence of extreme points

Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y_0\in Y$, the distance ...
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 ...
1
vote
2answers
477 views

Convex polynomial homogenization and convexity

I have a polynomial that I know to be convex. If I homogenize the polynomial, is the resulting homogeneous polynomial also convex? I know that the perspective of a convex function is convex, but ...
0
votes
0answers
216 views

Convexity and line segments

Let S be a subset of a linear space. Let S1 be the union of all line segments that join pairs of points in S. Now what happens if we repeat this process and construct S2, S3,....(Thus for example S2 ...
6
votes
1answer
354 views

Approximation of an integral of a concave function

I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) ...
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 ...
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 ...
3
votes
2answers
372 views

Submanifolds lying on the boundary of a convex domain

Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$. Is there any known condition that is equivalent to ...
2
votes
0answers
173 views

Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan. Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...
1
vote
0answers
340 views

Bounding point-wise maximum of the absolute difference of two convex functions

Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function. Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
17
votes
1answer
724 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, ...
5
votes
1answer
558 views

Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
1
vote
0answers
359 views

Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} ||x-y||=diam(K). $$ where $diam(K)$ denotes the diameter of $K$. ...
2
votes
2answers
325 views

If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces. I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex"). Thanks!
1
vote
0answers
126 views

Lower semicontinuity of Bregman distances/divergences

For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$: ...
3
votes
1answer
646 views

Lipschitz properties of minima/minimizers of convex functions of two variables

Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set $g(y) = \min_{x} f(x,y)$ What I would like is for $g(y)$ to be ...
3
votes
3answers
570 views

Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...
2
votes
1answer
269 views

Can subgradient infer convexity?

It is known that If $f:U→ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x_0$ in $U$ if for ...
4
votes
1answer
388 views

Polygons and mirrors

Given a point $A$ inside a non-convex polygon $P$, is it always possible to place a finite set of mirrors (not necessarily along the boundary of $P$, any position inside $P$ is allowed) given by ...
0
votes
1answer
284 views

Convex sets and projections

Hello! I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
5
votes
1answer
563 views

Orthogonal similarity of matrices

Given $M\in M_n({\mathbb R})$ and $\ell\in{0,\ldots,n-1}$, we define $$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$ where the indices are understood mod $n$. In particular, $d_0$ is the trace operator. Let ...
0
votes
3answers
521 views

a different algebra/representation for convex sets

Hi, I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...
8
votes
1answer
936 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 ...
11
votes
2answers
794 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 ...
5
votes
1answer
380 views

Convexity and Strong convexity of subsets of Surfaces

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly ...
2
votes
1answer
245 views

A differential inclusion relating to the slope of a convex function

This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...
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 ...
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? ...
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 ...
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 $P$, ...
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 ...
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 ...
4
votes
2answers
841 views

Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

Max Koecher (in, for example, The Minnesota Notes on Jordan Algebras and Their Applications (new edition: Springer Lecture Notes in Mathematics number 1710, 1999)), defined a domain of positivity for ...
1
vote
6answers
3k views

Extreme point compact convex set.

The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points. It seems this implies that a compact convex ...
5
votes
2answers
685 views

Maximal Ellipsoid

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
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 ...
4
votes
2answers
276 views

Efficiently sampling points from an integer lattice.

Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
4
votes
2answers
2k views

Compact Convex sets and Extreme Points

There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...
0
votes
2answers
2k views

If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...
4
votes
4answers
512 views

Radstrom cancellation only for two convex sets?

I've seen this statement of Radstrom cancellation: if $A +C \subset B+C$ where $A,B$ are convex, $B$ is closed, and $C$ is bounded, then $A \subset B.$ Is it essential that $A$ be convex?
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 ...
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 $\delta$. ...
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. ...