14
votes
3answers
588 views

“Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ ...
2
votes
1answer
154 views

Upper bounds on the worst-case traveling salesman tours in the unit square

The paper [1] proves that, if we place $N$ points in the unit square, then the length $\ell$ of the euclidean TSP tour of those points must satisfy $$\ell \leq \sqrt{2N} + 7/4~~.$$ I'm wondering, can ...
3
votes
1answer
190 views

Riemannian metric on a space of “not-quite-smooth” (hyper)surfaces?

Apologies if the title's a bit vague; I'll try to explain everything below, and please let me know if I should clarify anything. I've been looking for a while at variational problems on polytopes. ...
2
votes
0answers
143 views

Honeycomb-type properties of the Delaunay triangulation and Voronoi diagram

Suppose I'd like to distribute a set of points $P=\lbrace p_1 ,\dots, p_n \rbrace$ in the unit square $S=[0,1]\times[0,1]$ to minimize a weighted sum of two things: 1) The average distance between a ...
4
votes
0answers
123 views

Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set. It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...
5
votes
1answer
257 views

Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself: Definition. Let $(X,d)$ be a ...
11
votes
3answers
472 views

finding the most-isolated point in a high-dimensional cube

I have a set of points {$x_1,\ldots,x_n$} located in the d-dimensional unit cube $[0,1]^d$. $n$ is about 1000 and $d$ is about 25. I'd like to find $\max_{\omega\in [0,1]^d}\min_{i=1,\ldots,n} ...
7
votes
2answers
345 views

More general form of inequality?

I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise. The lemma says that for any set of vectors in ...
8
votes
2answers
338 views

An extension of Gaussian Isoperimetry

The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
2
votes
1answer
214 views

Maximum distance of points in intersection of balls

Dear all, let $B_\delta(p):=\{x\in\mathbb{R}^d:||x-p||_2\leq \delta\}$ be a $d$-dimensional closed ball. Now I do not have one ball, but four: $B_{r_1}(p)$, $B_{r_2}(p)$, $B_{s_1}(q)$ and ...
3
votes
3answers
776 views

Minimum norm of convex hull

Dear all, I am currently stuck at a problem which seems too easy to be stuck at to me... Summary Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute ...
13
votes
1answer
2k views

A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...
6
votes
3answers
407 views

Sequences of evenly-distributed points in a product of intervals

Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distributed" in a way which ...
5
votes
1answer
592 views

Minimizing variance of distances between points when mean distance is fixed

In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d ...
4
votes
0answers
280 views

The Gömböc and monostatic objects

This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I ...
3
votes
2answers
280 views

Optimizing the layout of Infinite Suburbia

Infinite Suburbia is a Euclidean plane, P. All residents live in open unit disks which, like caravans, can travel around but are stationary most of the time. When stationary, these disks lie in a ...
9
votes
2answers
2k views

An optimization problem for points on the sphere (master's dissertation)

First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
21
votes
2answers
1k views

What is the best way to peel fruit?

A mango made me wonder about this. (See also this question, which is in a similar spirit.) Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset ...
2
votes
1answer
96 views

Optimizing finite-length approximations to space-filling loops

Take a loop in the unit disk D^2, with length l where length is defined as the supremum of the lengths of piecewise linear approximations. What is the smallest r such that every radius-r subdisk of ...
2
votes
2answers
538 views

Are Bregman divergences quasi-convex?

Given a convex set S ⊂ ℝd and an appropriately differentiable convex function f: S → ℝ, a Bregman divergence Bf(x, y) = f (x) - f (y) -⟨x- y , ∇f (y)⟩ for x, y ...
3
votes
1answer
1k views

Maximizing Sparsity in l1 Minimization?

Consider the optimization problem $$\min_x ||Ax||_1 + \lambda||x-b||^2,$$ where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is ...
5
votes
3answers
602 views

How can I embed an N-points metric space to a hypercube with low distortion?

I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
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 ...