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Questions tagged [convex-geometry]

Use this tag when posting questions related to the concept of convexity and geometry. For example, for convex polygons.

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Computing projection onto the following closed convex set

Let $\mathbf{S}^n$ denote the space of symmetric, real-valued $n \times n$ matrices. Consider the closed convex set $$ \mathcal{C} := \{(X, x) \in \mathbf{S}^n \times \mathbf{R}^n : X \succeq xx^T,...
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Well-definedness of greatest convex function smaller than function

A proof I am reading relies on the existence in $\mathbb{R}^d$ of a greatest convex function $f_0:\mathbb{R}^d \rightarrow \mathbb{R} \cup \{-\infty\}$ less than or equal to a (continuous) function $f:...
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25 views

Is the Legendre transformation continuous?

Is the Legendre transformation continuous on the space of the convex $\mathbb R^{n} \to \mathbb R$ functions for the topology of the simple convergence? First the limit of a sequence of convex ...
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Polyhedra, half spaces and compact

I would like to show that $\{ x ; \forall j \in \{1,...,n\}, \langle x, z_{j} \rangle \le s_{j} \} $ is compact when the convex cone generated by the $(z_{j})_{1 \le j \le n}$ is $R^{d}$. I have been ...
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1answer
34 views

hyperbolic hexagon construction [on hold]

Can someone show a proof of the existence of a hyperbolic right angled regular hexagon? How would I make a decoration of $4n$ of these hexagons in a way where it glues the diagram for a hyperbolic ...
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24 views

Separation of two disjoint convex closed sets

Assume that $A,B\subset \mathbb{R}^n$ are two disjoint closed convex sets. Without using that $A$ and $B$ are closed sets, it follows already, that there is a non zero element $v$ and a real number $c$...
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In spectrahedra, are minimal rank points always extreme points?

Consider a matrix $A$ in a spectrahedron $S$ such that $$\mbox{rank}(A) \leq \mbox{rank}(B)$$ for all $B\in S$ and assume that at least one matrix $C \in S$ we have that $\mbox{rank}(A) < \mbox{...
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Maximal enclosed D-simplex

Is it possible to construct convex D-polyhedron $P$ that there exists a D-simplex $A$ spanned on vertices of $P$, that its ...
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96 views

Distance between a cone and a disjoint hyperplane

I seek to prove the following, which I guess is true: Define $A:=\{x \ge 0\} \subset \mathbb{R}^m$ and assume that $U\subset \mathbb{R}^m$ is an affine subspace with $A \cap U=\emptyset$. Show that ...
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1answer
46 views

Bounding a convex set by ellipses

Given any compact convex set $K$ in $\mathbb{R}^d$ with non-empty interior, does there exists an affine transformation $T$ such that: $$\overline{B} (0, 1) \subset T(K) \subset d \cdot \overline{B} (...
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Does a convex set look the same near its face?

Recall that a face of a convex set $X\subseteq\mathbb{R}^n$ is a convex subset $F$ of $X$ such that every line segment with endpoints in $X$ whose relative interior meets $F$ lies entirely in $F$. ...
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How did Steiner prove his famous formula?

In convex integral geometry and geometric measure theory, Steiner's formula is the name of the following elegant result: Let $B_n$ be the unit ball in $\mathbf R^n$. If $S$ is a nonempty bounded ...
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1answer
26 views

Euler's Formula To Show 2E=3V

I have this question: "Consider a convex polyhedron, all of whose faces are square or regular pentagons. Say there are m squares and n pentagons. Assume that each vertex lies on exactly 3 edges" ...
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Find a bound of the biggest radius of hypersphere in convex hull centered on centroid

Setting: We have a probability distribution on a space $\mathcal{X} \subset \mathbb{R}^d$, called $\rho(x)$, and we are given a sample of iid points $S^n = \{x_i\}_{i=1}^n$ from $\rho$. Let $K: \...
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23 views

Is an open midpoint convex set in $\mathbb{R}^n$ always convex?

It can be easily shown that a closed midpoint convex set in $\mathbb{R}^n$ is always convex, but it has occurred to me that the counter-examples showing a midpoint convex set may not be convex, are ...
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28 views

$conv\{\bar{x},\bar{y},\bar{z} \} \cap conv\{\bar{x},\bar{y},\bar{t} \} \cap conv\{\bar{x},\bar{z},\bar{t} \} = \{\bar{x} \}$

Let $\{\bar{x},\bar{y},\bar{z},\bar{t}\}$ be 4 points in $\mathbb{R^2}$, such that $conv\{\bar{x},\bar{y},\bar{z} \} \cap conv\{\bar{x},\bar{y},\bar{t} \} \cap conv\{\bar{x},\bar{z},\bar{t} \} = \{\...
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Distance on the sphere is convex

Let $\mathbb{S}^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:\mathbb{S}^n \to [0, \infty)$ defined by $d(p) = d(p,p_0) = \cos^{-1}( \langle p, p_0 \rangle)$. It ...
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23 views

Prove that this set of expected values is convex

I am facing the problem of proving that the following set $K$ is convex, it's part of a proof for the First Fundamental Theorem of Asset Pricing: $$K := \{d \cdot E_\mathbb{Q}(\boldsymbol{X}): \...
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1answer
65 views

For which $C\subset \Bbb R^n_{++}$, the set $\log(C)$ is convex?

Let $\Bbb R^n_{++}=\{x\in\Bbb R^n\mid x_i>0, i=1,\ldots,n\}$ and let $\log\colon \Bbb R^n_{++}\to \Bbb R^n$ be the component-wise logarithmic function, i.e. $$\log(x)=(\log(x_1),\ldots,\log(x_n))\...
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sum of convex cone with apex at 0 and its closure

Let $K\subseteq\mathbb R^n$ be an open convex cone with apex at 0, and let $\mathrm {cl}K$ be its closure, how to prove that $K+\mathrm {cl}K=K?$
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How to understand the Minkowski content? [duplicate]

Picture below is from https://en.wikipedia.org/wiki/Minkowski_content . Although I can the word, but I really don't know what is upper Minkowski content and lower Minkowski content , how to ...
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Measure convex hull point and ball

The Lebesgue's measure (in $R^{n}$) of the convex hull of $B(0,a)$ and a point $p$ is $C_{n} a^{n-1}|p|$ with $C_{n}$ constant depending only on the dimension $n$. I don't know how to prove it, I ...
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21 views

Cone in higher dimensions

Let $\Omega$ be a bounded open convex set of $R^{n}$, $u \in C^{0}(\bar{\Omega})$ a convex function, and $v$ a convex function whose graph is the upside down cone with vertex $(x_{0},u(x_{0}))$ and ...
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Finitely generated cone definition clarification

The definition for a finetley generated cone says that a set $C$ is a finetly generated cone if there exists {$x^{(1)},x^{(2)},...,x^{(k)}$} such that $C$ = {$\sum\lambda_ix^{(i)}, \lambda \geq 0$}. ...
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1answer
37 views

Intuition behind existence of mixed volumes?

Consider "Volume" as a function from set of $d$-dimensional convex bodies to real numbers. This function is homogeneous of degree $d$ (under rescalings of the convex bodies). Minkowski's theorem ...
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1answer
42 views

Definition of convexity via probability measure

I found this alternative definition of convexity in a set of lecture notes on convex optimization: A subset $K \subset \mathbb R^n$ is convex if and only if for every probability measure $\mathbb P$ ...
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1answer
22 views

How to prove $||P_\Omega(v)-P_\Omega(u)|| \le ||v-u||, \forall u,v \in \mathbb{R}^n$?

Suppose $\Omega$ is a closed convex set in $\mathbb{R}^n$. Let $P_\Omega(u)$ represent the projection of $u$ onto $\Omega, \forall u \in \mathbb{R}^n$. That is to say $P_\Omega(u)= \underset{v \in\...
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Given two polytopes $P$ and $Q$, show that $(P^* \times Q^*)^* = P \oplus Q $

Using definitions, I got so far as to express $(P^* \times Q^*)^*$ in the following form: $$(P^* \times Q^*)^* = \left\{\,\begin{pmatrix} z^* \\ w^* \end{pmatrix} \in \mathbb{R}^{d+e} \,\middle|\,...
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1answer
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Uniform sampling of convex polytopes: why is it hard?

There is a surprising number of posts asking about the uniform sampling of convex polytopes (e.g. 1, 2, 3), and an equally surprising number of non-answers (e.g. a, b). From what I have read, this ...
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1answer
45 views

Separating hyperplane

Let $K_1,K_2$ be disjoint convex sets in $\mathbb C$. Let $z_1\in\partial K_1,z_2\in\partial K_2$ be the minimizers of $\mbox{dist}(K_1,K_2)=\inf{|x-y|}$ where $x\in K_1$ and $y\in K_2$. Is it true ...
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Basic question about definition of random polytopes

I am trying to read about random polytopes, the convex hull of n random points $x_1,\ldots, x_n$ chosen independently inside a convex body $K$ with respect to uniformly distribution. My questions are: ...
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Volume-minimizing symmetric convex body of constant width

If $K$ is any symmetric convex body with ${\rm vol}(K)={\rm vol}(B_1(0))$ where $B_1(0)$ is the unit ball in $\mathbb R^n$, do we always have ${\rm width}(K)\leq{\rm width}(B_1(0))$?
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Finding a counterexample to a claim about nonnegative polynomials

A sketch of the proof of the following statement is in Barvinok's "A Course in Convexity". Fixing positive integers $k$ and $n$, define $H_{2k,n}$ to be the real vector space of all homogeneous ...
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Classifying Radon partitions in $\mathbb{R}^n$ whose affine hull is $\mathbb{R}^n$

Specifically, I want to determine all distinct "types" of Radon partitions of $n+2$ points in $\mathbb{R}^n$ for which the affine hull is all of $\mathbb{R}^n$. This is a homework question, so I'm ...
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1answer
28 views

Proof of Kirchberger's Theorem in Convex Geometry

I am having trouble understanding certain parts of the proof of Kirchberger's Theorem, as presented here. Specifically, I am having problems understanding the proof of the following combinatorical ...
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Convex sets of width 1 have segments of length 1 in every direction

A strip of width w is a part of the plane bounded by two parallel lines at distance $w$. The width of a set $X \subseteq \mathbb{R}^2$ is the smallest width of a strip containing $X$. Prove that a ...
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1answer
63 views

Finding the convex hull of m x n matrices

Given a set of $\mathbb{R}^{m \times n}$ matrices, I would like to find the matrices forming the vertices of their convex hull. Would this be the same problem as finding the convex hull of a set of ...
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56 views

Intersection of any 2 intervals $\implies$ all intervals intersect

Let $\mathscr{F}$ be a finite family of open or closed intervals in the line $\mathbb{R}^1$. Show by an elementary proof (without referring to Helly's theorem), that if any $2$ of the intervals ...
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26 views

A more efficient convex hull algorithm

Before I start, I would like to say that this is for a programming project of mine I'm doing but I figured my question is only about the part involving math so here I am. So in Grahams Scan algorithm, ...
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Mapping between two sets of half spaces

Consider a non-empty, bounded $\mathcal H$-polyhedron described by $$P = \left\{ x \in \mathbb{R}^n : Ax \leq b \right\}$$ Now I have a linear transformation, $y = B x$, where $B \in \mathbb{R}^{m\...
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1answer
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Affine space and convex sets in the context of Euclidean space

I am a bit confused as to the relationship between the ideas of vector space, affine space, and convex sets in the context of Euclidean space $\mathbb{R}^d$. As of now, this is how I see it. $\mathbb{...
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Always a hyperplane through the origin between a point and a convex cone

Let $C$ be a closed convex cone in $\mathbb{R}^d$, that is, a closed convex set $C\subseteq \mathbb{R}^d$, so that for every $x\in C$ and for every $\lambda \ge 0$, the point $\lambda x$ is also ...
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When is the measure of spherical cap large?

It is known that in high dimensions, the measure of the spherical cap is small, due to the measure concentration for the sphere. In particular, we have the following inequality in $n$ dimension: $$ 1-\...
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Proof that makes use of the differentiability of a function and of its convex conjugate

I would like your help to understand what are the crucial assumptions driving the claim reported below. Let me start with the notation $\mathcal{Y}\equiv \{1,2,...,M\}$. $S$ is a random vector with ...
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1answer
21 views

Asymptotic behaviour of the intersection volume of balls with the same radius

Let $x,y \in \mathbb{R}^n$ be two fixed points. Is there an easy proof of the fact that $$A(r):=\frac{ \text{Vol}(B(x,r) \cap B(y,r))}{\text{Vol}(B(x,r))}$$ tends to $1$ when $r \to \infty$. I ...
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29 views

What is the difference between convex cone and convex hull?

I am reading this definition of convex cone and this definition of the convex hull of a finite set of points and I am in trouble in understanding the difference. Am I right that, given a set of ...
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1answer
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What does the set : $\{ (\lambda_1, …, \lambda_n) \in \mathbb{R}_+^n \mid \sum \lambda_i = 1 \}$ represent

What the set : $$S = \{ (\lambda_1, ..., \lambda_n) \in \mathbb{R}_+^n \mid \sum \lambda_i = 1 \}$$ represent geometrically ? I tried in dimension $2$ and it seems that I get a triangle. But I ...
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Why is the epigraph of Moreau-Yosida Regularization a projection of a convex set?

The Moreau-Yosida Regularization is given by \begin{equation} f_\mu(x) = \inf_y \left( f(y) + \frac{1}{2\mu} \| x - y \|^2 \right). \end{equation} We know that $L(x, y) = f(y) + \frac{1}{2\mu} \| x -...
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1answer
64 views

Algorithms for projecting a point onto the convex hull spanned by a set of vectors

Given a set of vectors $V = \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset \mathbb{R}^d$, I want to project a point $\mathbf{x}_0 \in \mathbb{R}^d$ onto the convex hull $\text{conv}(V)$ of the ...
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67 views

Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?

I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a generic system of Laurent polynomials is the mixed volume of the polynomials' Newton ...