# Tagged Questions

A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

217 views

### does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$. I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...
57 views

### A question about the approximation of convex cones

I have the following question which maybe is too naive. Let $K$ be a convex cone on $\mathbf{R}^n$. Can we approximate $K$ by a sequence of polyhedral convex cone $K_i$ such that for any compactly ...
135 views

### Bang's open question strengthening Tarski's planks problem

Tarski's Planks problem, solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires "planks" (parallel strips) of total width $\ge d$ in order to completely cover a ...
83 views

36 views

138 views

147 views

90 views

### Diameter of a convex body relative to its Legendre ellipsoid

Given a convex body in $\mathbb{R}^n$ that is symmetric with respect to the origin, let us measure its diameter with respect to the Euclidean metric determined by its own Legendre ellipsoid. How large ...
127 views

### Geodesics on convex hypersufaces

Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior. Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest path)...
113 views

### polynomial expression for counting number of integral points of a set

Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$ Can we ...
145 views

### Regions on a sphere that avoid a fixed point set

Let $P$ be a finite set of points on a unit-radius sphere $S$ in $\mathbb{R}^3$. Treat $P$ as a fixed pattern that can be rigidly slid around $S$ as a unit (no reflection). Let $R$ be a subset of $S$....
77 views

### Linear projections of convex sets with unique preimages of boundary points

Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P_S$ denote the orthogonal linear projection onto $S$. I'd like to claim ...
547 views

### Helly's theorem in other areas of mathematics

Are there some outstanding results using some version of Helly's theorem in a totally different area (whatever that means) than convex geometry?
46 views

### Bounds on the spherical measure of sub-level sets of quadratic forms

I'm wondering if there are any bounds on the spherical measure of sets of the form $$\mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha)$$ where $\alpha$ is some ...
91 views

### planes intersecting a convex polytope

We are given a $d$-dimensional convex polytope ${\cal P}$ in $N$-dimensional space where $d<N-1$. Consider several planes $P_i$ corresponding to inequalities $f_i(X)\ge 0$. We are given that each ...
72 views

146 views

### Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$. Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
76 views

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and $J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ,\... 0answers 48 views ### Projecting a convex partition onto a convex set Say that$X$and$Y$are two convex regions in the plane, and suppose that$X \subset Y$. Further suppose that$Y$is partitioned into disjoint convex subsets$Y_1 ,\dots, Y_n$. Is there a way of ... 2answers 232 views ### Cardinality of non-integer points in the translation of the Minkowski sum of convex hull. Let$\operatorname{conv}(a_1,\ldots,a_m)$denote the convex hull of$\{a_1,\ldots,a_m\}$. Let$\mathbb{Z}_+=\mathbb{N}\cup\{0\}$and$\mathbb{Q}_+$denotes the positive (inluding 0) rational numbers. ... 1answer 217 views ### Extreme points of convex hull of Minkowski sum [closed] Let$\operatorname{conv}(a_1,\ldots,a_m)$denote the convex hull of$\{a_1,\ldots,a_n\}$. Let$P = \operatorname{conv}(a_1,\ldots,a_p)$and$Q = \operatorname{conv}(b_1,\ldots,b_q)$be two convex sets ... 0answers 45 views ### Projection of a ray onto a random polytope Suppose$P$is a polytope formed by$p$(general) random planes in$\mathbb{R}^n$. We assume$p \asymp n$and$P$has a diameter$O(\sqrt{n})$. For any$x \in \mathbb{R}^n$, denote by$\operatorname{...
The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$. Below I illustrate with $\cal C$ an equilateral triangle. Two players, $A$ and $B$, alternate turns. At each turn they ...