Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [convex-geometry]

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

Quoting Wikipedia: In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, etc.

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Need clarification on if this is a convex set.

Please refer to the photo I have attached to this question. A set S is convex if and only if for all $\textbf{a},\textbf{b} \in S$, the point $\textbf{x} = c\textbf{a} + d\textbf{b}$ is also in S, provided that $c,d \geq 0$ and $c + d = 1$. The…
Ran Rajmirsen
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Can we find a line L and a closed convex set S such that $L \cap S = \varnothing$, but each plane containing L intersects S

Question: In $\mathbb{R}^{3}$, Can we find a line L and a closed convex set S with $S \cap L = \varnothing$ such that for each plane $\Pi$, $L \subseteq \Pi$ we have $\Pi \cap S \neq \emptyset$? How can we find these line $L$ and the closed convex…
TrItOs
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Every three noncollinear $x, y, z\in S,\exists! p\in S:\overline{xp}\text{ , }\overline{yp}\text{ , }\overline{zp} \in S \Longrightarrow| \ker(S) |=1$

Problem: Let $\displaystyle{S}$ be a subset of $\displaystyle{\mathbb{E}^{n}}$. Suppose that $\displaystyle{S}$ contains at least three points that are not collinear, and suppose that for every three noncollinear points $\displaystyle{x , y , z}$ in…
TrItOs
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What is the smallest ball where a convex body of given diameter belongs to?

Let the diameter of the convex body $K$ be $c.$ It's easy to see that if the convex body is symmetric, then the convex body belongs to a ball $B(y,c/2).$ This bound is sharp in every dimension. But what if the convex body is not symmetric? Are the…
Johan Aspegren
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Convex sets and extreme points

I am learning about convex sets and extreme points from a course on linear programming. I came across a theorem that states that every closed convex set has an extreme point if and only if it does not contain a line. I cannot understand what it…
statBeginner
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A disk without a point on its boundary is a convex set?

I trying to show that any homeomorphism $h:D^2\to D^2$, where $D^2 =\left\{x\in \mathbb{R}^2: |x|\leq 1\right\}$ takes $S^1$ to $S^1$ and ${D^2}^\circ$ to ${D^2}^\circ$(interior of a disk). Supposed that $a\in {D^2}^\circ$ and $h(a) \in S^1$. Then,…
Horned Sphere
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$k\geq n+1$ convex sets in $\mathbb{R}^n$

Assume that $A_1,A_2,\dots,A_{k}\subset\mathbb{R}^n$ are $k$ convex sets and $k\geq n+1$. Is it true that there exist two distinct points $\gamma_1,\gamma_2\in\mathbb{R}_{\geq 0}^k$ such that…
Levent
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Parameterisation of convex shapes

Is there a way to describe any 2D shape from a set of independent parameters $p_0$, $p_1$, ... such that it is always convex no matter what the parameters are? The parameters can be limited to a fixed range $[{min}_i, {max}_i]$. For example, you…
Timmmm
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The closure of a cone, and its dual

Boyd's book on convex optimization makes the following claim about cones and their duals- If $K$ is a cone, and its closure is pointed, then $K^*$ has nonempty interior. Any hints on how to prove this? From sketching a few pictures in $\mathbb{R}^2$…
btimar
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Is the Isotropic Constant Scaling Invariant?

The entries of the covariance matrix of a convex body $K$ are defined as \begin{equation} \label{last} (a_{ij}) = \frac{\int_K x_ix_j}{|K|} - \frac{\int_K x_i}{|K|}\frac{\int_K x_j}{|K|}. \end{equation} We define the isotropic constant of any convex…
Johan Aspegren
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How to prove that the following is a convex set: $[3,5]\times [1,2]\times \{1\} \subseteq \mathbb{R}^3 $

This might be very easy, but I can not see how to prove that: $[3,5]\times [1,2]\times \{1\} \subseteq \mathbb{R}^3 $ is a convex set. Thank you for your help.
ergch24
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Can a part of the spherical surface be convex?

In my opinion, the line between two arbitrary points on the surface of a sphere is never part of the surface (the line is inside of the sphere). Hence a part of the spherical surface can't be convex. But I have read it differently. E.g. here:…
Dex124
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Is this combination of nonconvex sets convex?

Suppose I have a compact set $\mathcal A\subset \mathbb R^n$ that is not convex, and denote $\mathcal B = \{y \in \mathbb R^p : \|y\|_\infty \leq 1\}$ the unit infinity norm ball. For $p \geq n+1$, is the following set convex? $$ \mathcal S =…
Y. S.
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compact, convex set has $0$-dimensional face

Let $K \subset \mathbb{R}^d$ be compact and convex. Consider a point $x \in K$ such that $\lVert x \rVert$ is maximized and let $H$ be the hyperplane with equation $\langle y, x \rangle = \lVert x \rVert$. Is it true that $H \cap K = \{ x \}$ i.e.…
blm
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How to show that these sets are convex?

Consider an $n$ by $n$ matrix $A$, as arbitrary points on the $\mathbb{E}^{n^2}$ Euclidean space. Let $S$, $S_+$, and $S_{++}$ denote respectively the sets of symmetric, positive definite, and positive semidefinite $n$ by $n$ matrices on…
nehaci
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