# Questions tagged [convex-geometry]

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

1,012
questions

2
votes

0
answers

36
views

### Busemann-Petty type problems on complex vector spaces [closed]

We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...

0
votes

0
answers

26
views

### Iterating partially-unconstrained optimization with projection

Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize
$$
\tag{(A)...

1
vote

0
answers

77
views

### Distance between two convex sets

Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...

0
votes

0
answers

48
views

### Does convexity of boundary implies geodesic convexity?

I came across the following result (mentioned on Pg. 3 of this talk) that states that
If $D$ is an open connected subset of a complete Riemannian manifold with smooth metric then $\partial D$ convex ...

2
votes

1
answer

155
views

### Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...

0
votes

0
answers

24
views

### '$\alpha$-moments' and '$\alpha$-centers' of planar convex regions

We try to proceed from Least area and least perimeter triangles that contain a convex planar region - how different can they be?
The partial answer given to the above question shows a convex ...

2
votes

0
answers

137
views

### How many unit cubes are needed to 'hide' a unit cube fully in 3D?

Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...

0
votes

0
answers

97
views

### On special points within convex solids with all planar sections passing through them having equal area

Question: If within a convex solid body C there is a special point P such that every planar section of C passing through P has the same area, then, can we assert that C is a sphere and P its center? ...

1
vote

0
answers

29
views

### Variants of cutting plane method for convex optimization

The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...

0
votes

0
answers

28
views

### Set of enclosed convex polyhedra in a graph

Given a straight-line graph embedded in $\mathbb{R}^3$ with known vertex coordinates and edges and no edge intersections, is it possible to find all the enclosed convex polyhedra inside? If so, is ...

1
vote

1
answer

77
views

### Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...

0
votes

0
answers

21
views

### Directions of differentiability of log-concave measures with infinite-dimensional support

I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...

0
votes

1
answer

110
views

### Centroid of Minkowski sum

Let $A$ and $B$ be two compact convex subsets of $\mathbb{R}^n, n\geq 2$. Assume $x_A$ and $x_B$ are their respective centroid. If we form the Minkowski sum $C=A+B = \{x+y\mid x\in A, y\in B\}$, what ...

3
votes

0
answers

36
views

### Implementation of Friedman's algorithm of reconstructing simple polytopes

In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...

0
votes

1
answer

96
views

### Convex sets via fixed point equations

I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...

1
vote

0
answers

40
views

### Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces

Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...

1
vote

0
answers

83
views

### Polyhedrons with mutually non-congruent faces, all of equal area

This question is closely related to Convex polyhedra with non-congruent faces
It is known that if all faces of a tetrahedron ought to have same area (or same perimeter), then, the faces are ...

1
vote

1
answer

168
views

### Van der Waerden conjecture and Alexandrov-Fenchel inequality

The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel ...

7
votes

3
answers

625
views

### A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...

1
vote

1
answer

88
views

### To place copies of a planar convex region such that number of 'contacts' among them is maximized

A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...

4
votes

1
answer

181
views

### What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...

0
votes

0
answers

47
views

### Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?

By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.
Question: For specified ...

6
votes

1
answer

354
views

### Is the gradient of a strictly convex, continuously differentiable function a homeomorphism?

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $f:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.
$\nabla f$ is a bijection, but is ...

4
votes

1
answer

130
views

### Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...

2
votes

1
answer

198
views

### Prove inequality: $2\Big[ \sum_{k=0}^{n} (k+1) a_k \Big]^2 -\Big[1+ \sum_{k=0}^{n} a_k \Big]\Big[ \sum_{k=0}^{n} k(k+1)a_k \Big] \geq 0.$

Suppose $a_0\geq\dots \geq a_n \geq 0$ is a sequence of non-negative numbers, where $n+1\leq \sum_{k=0}^{n} a_k \leq n+2$. Then, I want to prove that the following statement is true,
$2\Big[ \sum_{k=0}...

1
vote

1
answer

59
views

### Joint maximizer of a strongly concave function

I have a question that is arising in my research.
Suppose that $f : \mathbb{R}^ 2 \to \mathbb{R}$ is a strongly concave function, satisfying:
For every $x$, the function $y \to f(x, y)$ is maximized ...

3
votes

0
answers

115
views

### Reference request: Carathéodory-type theorem for convex hulls of closed sets

I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...

0
votes

0
answers

35
views

### For a convex body $K\subset R^n$, does the quantity $\min_{t>0} E[t^{-1} \|tg - \pi_K( t g)\|_2]$ have a name? Where has it been studied?

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).
Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as
$$
\Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\|
$$...

1
vote

1
answer

317
views

### Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$.
Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$.
For ...

0
votes

1
answer

164
views

### Density of the set of convex polygons in the Banach-Mazur distance

Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!

4
votes

1
answer

166
views

### Convex hull of 3 points in Cartan-Hadamard manifolds

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...

2
votes

1
answer

103
views

### To find the convex planar region minimizing diameter when area and perimeter are given

The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...

0
votes

1
answer

123
views

### On 'special' points on uniform planar convex regions defined in terms of moment of inertia

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...

2
votes

1
answer

148
views

### Conic hull of a rectangle

I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...

2
votes

1
answer

134
views

### On the moment of inertia of planar convex regions and possible special nature of circular disks

We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference ...

1
vote

1
answer

90
views

### If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?

That is, if $P$ is a finite set, and $C$ is the set of points in $P$ which lie on the boundary of the convex hull of $P$, then is $P$ contained in the convex hull of $C$?
It seems true intuitively. In ...

1
vote

0
answers

16
views

### Parabolic equations and nonconvex domains

Is there a (system of) parabolic equation(s) where qualitative properties depend on whether or not the (say, smooth and bounded) domain $\Omega$ is convex? I am aware of a few cases where a proof ...

17
votes

0
answers

395
views

### Talagrand's "Creating convexity" conjecture

We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...

1
vote

0
answers

102
views

### Planar sections of convex sets in Cartan-Hadamard manifolds

Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...

2
votes

1
answer

126
views

### On equal area planar sections of 3D convex bodies

This is an extension of On segments of equal area cut from planar convex regions by chords.
While the 3D analog of the above question would be about 3D pieces cut from a convex body $C$ by planes, ...

2
votes

0
answers

136
views

### On segments of equal area cut from planar convex regions by chords

Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...

1
vote

0
answers

115
views

### Smooth action on cotangent space of the plane

Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by
$$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$
acts via ...

1
vote

0
answers

237
views

### A sort of dual to nondegenerate random variables

I was motivated by this classical puzzle/1992 Putnam problem.
Suppose 4 points are independently and uniformly distributed on a sphere in 3d. What is the probability the tetrahedron they form contains ...

11
votes

0
answers

664
views

### John-type theorems: trading structure for accuracy?

Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that
$$ B \subset TB' \subset \tau B$$
for ...

2
votes

0
answers

45
views

### Estimating the Hausdorff distance of parallel facets of convex polytopes

Background
Let $\mathcal{K}_P^n$ denote the class of open, convex, $n$-dimensional polytopes in $\mathbb{R}^n$ containing the origin. For each $K\in \mathcal{K}_P^n$, its gauge function $f_*:\mathbb{R}...

0
votes

0
answers

54
views

### Zero flux along lines

I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...

0
votes

0
answers

41
views

### 3d convex body with max kissing number with translates

Ref: Convex region $C$ with least kissing number of copies of $C$
Given a convex body $C$, let us define its 'translate kissing number' $k_t$ as the largest possible number of translated copies of $C$ ...

3
votes

1
answer

34
views

### A converse question about the polyhedrality under linear mapping

It is quite well known that for any polyhedral set $K$ and any linear mapping $A$, the set $AK$ is polyhedral and hence closed. I am more curious about the converse problem, namely:
Suppose $K$ is a ...

0
votes

0
answers

31
views

### Understanding $k$-rotundity

We know that the definitions of $k$-Uniform rotundity ($k$-UR) or locally $k$- uniform rotundity (L$k$-UR) act to find the volume in higher k-dimensional spaces by defining $V(x_1, ..., x_{k+1})$ (For ...

1
vote

1
answer

147
views

### Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand

I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...