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2
votes
1answer
267 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
377 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
555 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
514 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
890 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
756 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
372 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
243 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 ...
18
votes
5answers
9k 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
707 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
466 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
818 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
668 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
404 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
269 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
490 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
599 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. ...