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3
votes
0answers
113 views

Maximising the volume of a parallelotope in a cone with a fixed vertex

Let $C\subset\mathbb{R}^n$ be a cone with vertex at the origin, aperture $\theta$ and height $h$. Since a cone is a convex region in $\mathbb{R}^n$ we know there is a parallelotope $P$, completely ...
9
votes
2answers
470 views

Concentration of measure for arbitrary convex bodies?

There are various "concentration-of-measure" theorems, the best known that due to Lévy, which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely concentrated around an ...
5
votes
2answers
315 views

Distributing points with respect to a concave function

Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ ...
0
votes
0answers
197 views

How do control the boundary regularity of the Legendre transformation domain from a convex function

Let f(x) be a strongly convex smooth function (its Hessian matrix is positive definite) defined in a convex domain D, introduce the Legendre transformation $$x=(x_1,...,x_n)\rightarrow ...
4
votes
3answers
646 views

Area-preserving map between rectangles and fat polygons

Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle ...
10
votes
2answers
902 views

The cone of positive semidefinite matrices is self-dual? (reference needed)

I'm seeking a reference for the following fact. The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar). This result is relatively easy to prove, has been known for a long ...
6
votes
1answer
1k views

Geometric meaning of derivatives of the curvature

Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ ...
12
votes
3answers
595 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
3
votes
3answers
744 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 ...
2
votes
1answer
350 views

A min-max formula for depth of the origin in a convex set

Depth is defined as the distance to the boundary of a set, i.e., $\operatorname{depth}(x, C) = \operatorname{dist}(x, \mathbb{R}^n \backslash C)$. Let $C$ be a convex set that contains the origin. I ...
10
votes
2answers
556 views

Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$. Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$. It seems that not ...
3
votes
1answer
362 views

minimal maximal ellipsoids

Suppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If ...
6
votes
1answer
750 views

Equations for an algebraic gömböc

A gömböc is a $3-$dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c). Such a convex ...
3
votes
0answers
203 views

Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an element $C\in\mathcal C_d$ ...
8
votes
1answer
540 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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
3
votes
0answers
183 views

Maximum surface area among convex subsets of the unit sphere of a given volume

The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased): Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define ...
3
votes
2answers
237 views

volume of the projected body

Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp ...
35
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 ...
8
votes
3answers
950 views

Listing lattice points in a simplex

Let $n \in \mathbf{Z}_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v_0,\dots,v_n \in \mathbf{Z}^n$, computes the set $\Delta ...
9
votes
0answers
319 views

How aspherical can a Gömböc be?

A Gömböc is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the ...
30
votes
0answers
862 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
363 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 ...
16
votes
4answers
1k views

Analogy of Parseval identity for Legendre Transform ?

Parseval's identity states that the sum of squares of coefficients of the Fourier transform of a function equals the integral of the square of the function, or $$ \sum_{-\infty}^{\infty} |c_n|^2 = ...
12
votes
2answers
577 views

Point on a line nearest a point in Banach space

I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of ...
7
votes
3answers
490 views

sharper minkowski bound

If we want to bound the norm of the smallest ideal which generates a nontrivial ideal class, is there a better bound than Minkowski's bound? (Note that Minkowski's bound is to guarantee something ...
7
votes
2answers
466 views

Do elongated convex objects all have long simple geodesics?

Let $S$ be a closed convex surface, the boundary of a compact convex body in $\mathbb{R}^3$. I am interested in whether there are conditions on its shape that ensure that it supports a long, simple ...
2
votes
2answers
917 views

Finding points inside innermost convex hull [closed]

Given a set of points $S$ on the Euclidean plane, Onion Peeling determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in ...
1
vote
1answer
295 views

How do maximum norms relatively change in Euclidean translations

Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$ that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible linear transformation from ...
1
vote
1answer
181 views

crookedness of convex curves (milnor)

hello, I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3) A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either ...
1
vote
1answer
252 views

The number of hyperplanes determining the integer points of a polyhedron

This question is inspired by this one. Let $P \subset \mathbb R_{+}^n$ be a convex polyhedron whose complement in $\mathbb R_{+}^n$ has finite volume. Let $Int(P) = P \cap \mathbb N^n$. (For ...
21
votes
4answers
1k views

Ellipse naturally associated with a polygon

My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
1
vote
2answers
582 views

Is there always a parallelogram cross-section of parallelepiped contained in the smallest box

Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...
1
vote
0answers
354 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 $K$. ...
20
votes
2answers
1k views

The logic of convex sets

Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...
3
votes
1answer
348 views

A conjecture of parallelogram inside convex and central symmetric curve

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$. How to prove the conjecture that $\displaystyle ...
8
votes
1answer
188 views

Finding a point that lies in a majority of polytopes

Suppose I have $k$ $n$-dimensional polytopes $P_1,\ldots,P_k$, each explicitly specified as the intersection of a collection of hyperplanes. If there was a point $p \in \mathbb{R}^n$ that lay in the ...
2
votes
0answers
113 views

Quantifying that near a point on a smooth hypersurface, it looks like a tangent hyperplane

Suppose $C$ is a smoothly bounded convex body (*) in $\mathbb{R}^d$, and $p$ is a point on the boundary of $C$. Let $r>0$ and let $B(r)$ denote the ball of radius $r$ centered at $p$. Is it ...
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 ...
5
votes
2answers
353 views

Isodiametric hull

Let A be a convex compact set in the plane (with a piecewise smooth boundary, say). We want to `inflate' it in such a way that the diameter does not increase. More accurately, we are looking for ...
3
votes
0answers
120 views

Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A. Has ...
25
votes
3answers
2k views

Polar body of a convex body that avoids a lattice

Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of ...
3
votes
3answers
856 views

Estimate probability( 0 is in the convex hull of N random points ) ?

Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95 for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ? The application is nearest-neghbour ...
2
votes
1answer
614 views

Condition number for Ellipsoid method matrix

Hello, When using the ellipsoid method (for solving a linear program for example), the volume of the ellipsoid at each iteration is proven to decrease, and do so by at least a factor of $e^{1/2n}$. ...
11
votes
5answers
705 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 $P$, ...
7
votes
1answer
426 views

A variation of Minkowski sum

I have to work with the following variation of Minkowski sum: Let $\mathbb E$ be a Euclidean space and $K$ be a convex set in $\mathbb E\times \mathbb E$. Set $$K^+=\{\\,x+y\in\mathbb ...
11
votes
1answer
378 views

Pushing convex bodies together

Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies ...
10
votes
0answers
594 views

Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup: Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
16
votes
3answers
1k views

Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference. Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...
0
votes
1answer
1k views

Minimum enclosing rectangle of a convex polygon proof

Hi, I've been reading about the rotating calipers algorithm for solving the minimum-area enclosing rectangle problem. It relies on a theorem: The rectangle of minimum area enclosing a convex polygon ...
4
votes
1answer
192 views

Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets

Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha_2(A)))-\log(\mathrm{Vol}(\alpha_1(A)))$ where $\alpha_1,\alpha_2$ are two affine bijections such ...