The convex-geometry tag has no wiki summary.

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### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

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votes

**1**answer

39 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq ...

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**5**answers

1k views

### How to explain the concentration-of-measure phenomenon intuitively?

One way to phrase the
"concentration-of-measure"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
...

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**0**answers

111 views

### What is the name for this measure on a Minkowski plane (=R^2 with a norm)?

Suppose $(V,\|\cdot\|)$ is a normed 2-dimensional real vector space.
Choose an inner product on $V$ and let $\lambda$ be the resulting Lebesgue measure. Set
$$K=\sup\ \lambda(\{\alpha a+\beta b\ :\ ...

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votes

**1**answer

129 views

### partition of a convex set into squares

Let $P$ denote the perimeter function. It's not hard to prove that for any rectangle $R$ in $\mathbb{R}^2$, $R$ can be partitioned into a countable collection of squares $\{Q_k\}_{k=1}^{\infty}$ such ...

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votes

**1**answer

42 views

### Common density of a random Vector with dependent entries ($Z=(aX,bX)$) [closed]

Given a real random variable $X$ with density $g:R \to R$ and two real constants $a,b$.
Let $Z:=(aX, bX)$, the random vector with entries as written (same random variable $X$ with different factors).
...

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votes

**1**answer

92 views

### Lattice parallelogram of minimal area containing convex lattice polygon

What is the minimal constant $\alpha$ so that for any convex lattice polygon $F$ there exists a lattice parallelogram $P\supseteq F$ of area $A(P)\leq \alpha\cdot A(F)$?
It is not hard to show that ...

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**1**answer

129 views

### Measurement of “symmetry” of a convex body

I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or ...

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**4**answers

289 views

### When is the convex hull of two space curves the union of lines?

I am interested in the convex hull of two space curves. Let $A,B\subset \mathbb R^3$ be two spaces curves. I am interested in when $\mathrm {con} (A \cup B)$ equals to $$\bigcup _{a\in A,b\in B} ...

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39 views

### center of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...

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34 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 ...

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**2**answers

110 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 ...

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votes

**1**answer

104 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 ...

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**1**answer

107 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 ...

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**0**answers

46 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 ...

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votes

**3**answers

464 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?

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**0**answers

40 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 ...

**2**

votes

**1**answer

81 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 ...

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**0**answers

54 views

### Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial ...

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**2**answers

62 views

### Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...

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126 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 $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

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61 views

### Newton polyhedron and product of ideals

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 ...

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**0**answers

39 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 ...

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vote

**1**answer

173 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. ...

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vote

**1**answer

90 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 ...

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40 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 ...

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234 views

### A (possibly boring) Voronoi Game

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 ...

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**1**answer

179 views

### Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given ...

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**0**answers

82 views

### What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a product of trees? )
Fix an integer $k \ge 2$, and let
$MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ ...

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**1**answer

144 views

### Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse.
That is, given an $n$-dimensional ellipsoid ...

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**1**answer

259 views

### Radius of the largest enclosed ball in the convex hull of an algebraic variety

Let $\mathcal{V}\subset\mathbb{R}^n$ be a real compact algebraic variety. Let $\mathcal{V}^c$ be the convex hull of $\mathcal{V}$ and let us assume that $\mathcal{V}^c$ has nonzero n-dimensional ...

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36 views

### Averaging a log-concave centrally-symmetric function over convex bodies

the question is the following: given a log-concave isotropic function $f$ defined on $\mathbb{R}^n$ and two compact 0-symmetric convex bodies $C_1, C_2$, where $C_1 \subset C_2$. Is it true that
$$
...

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135 views

### Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...

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68 views

### On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m ...

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**0**answers

73 views

### Largest ball with fixed center in a a convex region

Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...

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**0**answers

111 views

### Intersecting balls with convex regions and a bisector thereof

This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...

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**1**answer

129 views

### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...

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**1**answer

409 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in ...

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87 views

### If $\angle 0xy\leq \pi/2$ for every $x\in \partial K$, $y\in K$, then $K$ is a ball

Let $K$ be a bounded closed convex set in $\mathbb{R}^d$, $0$ lies in interior of $K$. Assume that for every boundary point $x$ of $K$ the plane through $x$ perpendicular to $x$ is a support plane of ...

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**1**answer

161 views

### Upper bound for the number of integral points in a convex set

Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer
coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?

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**1**answer

119 views

### Approximation of convex body by polytopes

In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$,
$$\rho_H(U,\mathcal{P}_n)\leq c(U) ...

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**1**answer

130 views

### Convex hull of the union of two parameterized curves in $\mathbb{R}^3$

My goal is to find a way to calculate the convex hull of the union of some parameterized curves.
For instance, I had to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup ...

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59 views

### max volume of inscribed simplex in a ball

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\epsilon)$ be the spherical cap with height $\epsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in ...

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43 views

### Lower bound for Quermassintegrals

Let $K$ be a convex body (a compact, convex subset of $R^n$ with empty interior), denote $W_0(K),W_1(K),\cdots,W_n(K)$ the Quermassintegrals of $K$. Note that $W_0(K)=V(K)$ the volume of $K$, and ...

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85 views

### two-dimensional sections of polyhedral cones

Given a polyhedral cone, its intersection with any two-dimensional plane is either a polygon or a region enclosed by a polygonal curve. Is it a characterization of polyhedral cones? Does there exists ...

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128 views

### Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...

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27 views

### Looking for a homogeneous function with some properties

I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...

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32 views

### continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following:
Does a continuous and concave function
\begin{eqnarray*}
f: N_{\mathbb{Q}} \to \mathbb{R}
...

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50 views

### Integral representation formula for convex

For $u \in \mathbb{S}^{d-1} \subset \mathbb{R}^d$, it is easy to show that:
\begin{equation}
u=c_d \int_{\mathbb{S^{d-1}}} \xi \mathbb{1}_{\left\{x \cdot u >0 \right\}}(\xi) \ ...

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**1**answer

976 views

### Which natural numbers are a square minus a sum of two squares?

Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?
This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...