The convex-geometry tag has no usage guidance.

**4**

votes

**0**answers

42 views

### Convex hull of the orbit of a matrix under permutations

Let $P$ be a generic permutation matrix on $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, the convex hull of the set $\{ Px : \; \text{$P$ is a permutation matrix}\}$ is the set of vectors ...

**4**

votes

**1**answer

120 views

### Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...

**5**

votes

**0**answers

65 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**0**

votes

**0**answers

20 views

### Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...

**1**

vote

**0**answers

40 views

### Number of simplices contained in a convex body

I am interested in the following question:
Given a convex body $K$ in $\mathbb{R}^d$ and an $\epsilon>0$ small enough, how many $d$-simplices $\{D_i\}_{i=1}^m\subset K$ do we need at least so that ...

**0**

votes

**0**answers

50 views

### Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that ...

**13**

votes

**0**answers

214 views

### A functional inequality about log-concave functions

Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle ...

**1**

vote

**1**answer

70 views

### Internal edges in Convex Polytopes

Suppose $S\subset{\mathbb R}^n$ is an infinite subset that is in general position which means that the intersection of $S$ with every affine subspace of dimension $d<n$ always contains at most ...

**4**

votes

**0**answers

90 views

### Alexandrov-Fenchel inequality for sets of positive reach

If $E$ is a convex subset of $\mathbb{R}^n$ with $|E| = |B_1|$, then one consequence of the classical Alexandrov-Fenchel inequalities from convex geometry is that
$$\int_{\partial E} H_{\partial E} ...

**1**

vote

**0**answers

51 views

### Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb ...

**0**

votes

**1**answer

96 views

### Convex cones: strict separation

Consider two closed convex cones $A$ and $B$ in $\mathbb{R}^3$. Assume that they are convex even without zero vector, i.e. $A \setminus \{0\}$ and $B \setminus \{0\}$ are also convex (it helps to ...

**4**

votes

**0**answers

171 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...

**4**

votes

**1**answer

140 views

### Ascertain properties of a new kind of rectilinear-convex set

PREABMLE TO MY QUESTION
I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many ...

**3**

votes

**1**answer

127 views

### A family of convex bodies in Banach-Mazur position

Let $\{K_i\}$ be a family of smooth, origin-symmetric, strictly convex bodies such that $K_i$ converge in the Hausdorff distance (or you may assume $\partial K_i\to \partial K$ smoothly, in the sense ...

**0**

votes

**0**answers

93 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 ...

**0**

votes

**0**answers

39 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 ...

**7**

votes

**0**answers

107 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 ...

**5**

votes

**1**answer

69 views

### Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...

**2**

votes

**1**answer

99 views

### Set of regular points in an Alexandrov space with curvature bounded below

Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$.
...

**1**

vote

**0**answers

51 views

### Finding optimal linear transformation for intersection of convex polytopes

I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in ...

**3**

votes

**1**answer

153 views

### Matching moments in even dimensions

Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu_i=\mathbb{E}_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. ...

**6**

votes

**1**answer

153 views

### Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...

**5**

votes

**2**answers

97 views

### Inscribed parallelotope in a $d$-simplex

The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we construct a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N ...

**1**

vote

**0**answers

34 views

### Perimeters of the cells of a convex tessellation

Let $C$ be a compact, convex region in $\mathbb{R}^2$, and say we have scalars $a_i, b_i, c_i$ for $i\in\{1,\dots,n\}$. Consider the tessellation $R_1,\dots,R_n$ of $C$ defined by letting ...

**1**

vote

**0**answers

39 views

### Inscribed polytopal approximation to a convex body

This question is on the continuation of the post
Approximation of convex body by polytopes
The central problem I am interested is an explicit construction of inscribed polytope with at most $n$ ...

**3**

votes

**1**answer

183 views

### Find the minimum distance between two convex hulls

We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...

**5**

votes

**0**answers

56 views

### Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq ...

**3**

votes

**1**answer

96 views

### The sign of the mean curvature on convex cones in three dimensions

my question is as follows. It is known that a closed smooth curve in $R^2$ is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in $R^3$ in a ...

**0**

votes

**0**answers

55 views

### lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$.
What is the lattice ...

**4**

votes

**0**answers

89 views

### 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 ...

**7**

votes

**4**answers

108 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{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...

**17**

votes

**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
...

**5**

votes

**1**answer

155 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 ...

**6**

votes

**1**answer

108 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 ...

**5**

votes

**1**answer

166 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 ...

**9**

votes

**4**answers

328 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} ...

**0**

votes

**0**answers

50 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 ...

**3**

votes

**1**answer

80 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 ...

**6**

votes

**2**answers

126 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 ...

**0**

votes

**1**answer

112 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 ...

**7**

votes

**1**answer

139 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 ...

**4**

votes

**0**answers

65 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 ...

**7**

votes

**3**answers

499 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?

**1**

vote

**0**answers

44 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

89 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 ...

**5**

votes

**0**answers

65 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 ...

**4**

votes

**2**answers

70 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 ...

**2**

votes

**0**answers

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

**2**

votes

**0**answers

71 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 ...

**1**

vote

**0**answers

45 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 ...