# Questions tagged [convex-geometry]

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

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### Convex functions in convex sets

Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...

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### Understanding a claim of Makai and Martini: why is an ellipsoid's cross-section body the same as its projection body?

In The Cross-Section Body, Plane Sections of Convex
Bodies and Approximation of Convex Bodies, I (Makai and Martini, 1996), the authors define for a convex body $K$ its cross-section body $CK$ and its ...

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### Sliding a convex body over a Gaussian measure

Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density
$$
\gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}.
...

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### Representation of concave point-to-set maps

Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...

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### Closed form volumes for intersecting modified cylinders

This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...

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### A question on strongly convex rational polyhedral cone

Let A be a strongly convex rational polyhedral cone in R^n.
Does (A+(-A))⋂Z^n=(A⋂Z^n)+(-A⋂Z^n) hold?
If it holds, why does it hold?.

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### problem：Conditions for including cones [duplicate]

I have considered a very interesting question myself, and I think it is very difficult to answer it.
Consider N n-dimensional vectors, where the angle between any two vectors is acute and their ...

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### Extreme points of a two-dimensional convex body in terms of its surface area measure

Let $K \subset \mathbb{R}^2$ be a nonempty compact convex set.
For any $t \in S^1$, define the unit vector $u_t = (\cos t, \sin t)$ making an angle of $t$, and let $l_K(t)$ be the tangent line of $K$ ...

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### Conditions for including cones

Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...

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### Tangent cone of a closed convex cone

Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...

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### good textbooks which deal with convex geometry, especially convex polyhedral cones and convex polytopes

I study toric varieties. In toric geometry, we use convex geometry, especially convex polyhedral cones and convex polytopes. Are there good textbooks which deal with convex geometry, especially convex ...

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### An upper bound of gradient norm for convex functions near minimizer

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...

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### Is the interior of the tensor product of two convex cones equal to the tensor product of their respective interiors?

I am sorry that the following question is elementary. I have not received an answer from my post at Math Stack Exchange.
In the following question, all cones are convex and contain the origin. Let $C \...

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### Number of cliques of intersection graphs in hyperconvex metric spaces

(cross-posted from Math.SE)
I would like to find out if it is known whether intersection graphs of closed unit balls in hyperconvex metric spaces have polynomially many cliques.
A metric space is ...

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### Expectation of dual norm induced by probability measure

Let $\mu$ be a probability measure supported on the unit sphere $\mathbb{S}^{d-1}$. Assuming that $\mu$ is even and not supported on any great subsphere, its cosine transform $w \in \mathbb{R}^d \...

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### Universal covering problem in dimension three

In dimension two one can quickly prove that every shape of diameter $1$ can be covered with a regular hexagon of width $1$. Thus, one may ask what is the minimal area of such a "universal ...

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### Quantifying error in the reconstruction of convex polytopes from moments

The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...

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1
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### Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...

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### Partitioning convex regions, maximizing the average perimeter of pieces

We continue from Cutting convex regions into equal diameter and equal least width pieces - 2
Question: If a planar convex region C is to be cut into n convex pieces such that the average of the ...

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### Fitting a convex polytope with 𝑛 facets between two nested spheres

This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...

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### Inside-out dissections of polygons - a generalization

Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...

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### The bounded complex of a polyhedral decomposition

Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
The union ...

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### On convexity of special fractals in the plane

Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.
For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...

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### Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

I have difficulty even in finding a Russian version of the next paper:
"Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex
function and some properties of ...

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### Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?

Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...

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### Maximum number of vectors with bounds on inner products (follow up question)

This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...

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### Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?

Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function.
Let $X$ be log-concave ...

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### Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...

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### The closest ellipse and circle to a given triangle - 2

We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...

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### Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...

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### Toric decomposition of multipartitions

Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...

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### Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...

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### abstract description of the topology on a real vector space defined by the algebraically open sets

Let $V$ be a real vector space. Given a subset $A \subseteq V$, say that a point $x \in A$ lies in the algebraic interior of $A$ if every affine line $\ell$ that passes through $x$ has the property ...

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### On equipartitions of surfaces of 3D convex regions

Let S be the surface of a 3D convex region (a 'convex surface'). Let S' be a subset of S. We shall refer to S' as geodetically convex in S if the following condition holds: If A and B are two points ...

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### Can a convex frame hold all circles of radius $1/n$ immobile?

Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile.
By "immobile", I mean no circle can move without overlapping other circles or the frame, ...

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### On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...

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### Smallest centrally symmetric containers of planar regions - 2

This post adds a bit to Finding the smallest centrally symmetric region that contains a convex planar region . In his answer there, Jukka Kohonen observed: Given a planar convex region $C$, the ...

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### 'Constrained morphing' of planar convex regions

Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...

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### Possible extensions of the perpendicular axes theorem for moment of inertia

This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...

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### Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia

Ref: Mathematical Omnibus by Fuchs and Tabachnikov, Lecture 11.
Consider any planar convex region C. A line l may be called an inertia bisector of C if it divides C into 2 pieces each of which has ...

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### Question about symmetric bilinear form and convex geometry

Consider a Finsler norm $\varphi$ in $R^4$, and a positive definite symmetric bilinear form $S$, there also exists a vector-valued symmetric linear form $Q$, such that if $S(v, w) = 0$, then $Q(v, w)$ ...

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### Geometric inequality related with convexity of the boundary

I'm new to Mathoverflow, so hopefully my question is well-posed.
My problem states as follows:
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...

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### Number of tiles inside a region of a hyperbolic tiling

Let $\mathbb{D}$ be the Poincare disc model of hyperbolic geometry with $\{p,q\}$ tiling on it.
In this, paper authors calculated the number of tiles on any circular region centered at a vertex or ...

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### Spherically convex set projection convexity

Let $K \subset S^n(0,1)$ be a geodesically convex set on a unit sphere, $x_0 \in \bar{K} \cap S^n(0,1)$ be some point outside of $K$ on sphere and let $X$ be a cartesian to spherical coordinates map, $...

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### On moments of inertia of planar and 3D convex bodies

The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one ...

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### Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...

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### What is the volume of a soft octatwister?

A soft octatwister is one of the 36 regular soft polytwisters, and is, up to similitude, the convex hull of a set of six circles in $\mathbb{R}^4$, with each circle defined by one of the following ...

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### Do subgradient inequalities hold for matrix convex functions?

Suppose $f$ is a matrix convex function over symmetric, positive semidefinite matrices with spectra in some interval $I$ [1]. That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$,
...

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### Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?

Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...

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### John and Lowner ellipsoid

I am looking at a proof to show that Lowner ellipsoids are unique for centrally symmetric convex body $K$. I want to show basically that $$Low(K)=John(K^{\circ})^{\circ},$$ where the $\circ$ means ...