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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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Proof of Separating Axis Theorem for polygons

One well-known special case of the Hyperplane Separation Theorem (Separating Axis Theorem) states: Let $A, B \subset \mathbb{R}^2$ be convex polygons. Then $A \cap B = \emptyset$ if and only if there is a straight line $L$, such that $A \subset…
ThomasR
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Complementary, convex sets in $\Bbb R^n$

Is it true or false? If $A\subset \Bbb R^n$ is convex and $\Bbb R^n\setminus A$ is convex then $A=\emptyset$, $A=\Bbb R^n$ or $\partial A$ is a hyperplane. I remember vaguely this fact from university, but I'm not sure if it is true or not. (My…
ajotatxe
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What exactly does a compact convex set mean.?

Let $A$ be a compact convex set in $\mathbb{R}^2$. What does ''compact convex set'' mean? What I understand: We have a "bunch" of real points $(x,y)$ in the plane. Any two of them satisfies the fact that a line drawn between them is fully inside…
This is not me.
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Convexity of trace$(X^{-1})$

Prove that the function $$ f(X) = \operatorname{trace}(X^{−1}) $$ is convex on the domain $S^n_{++}$. I was given the hint to try using line restriction. So I am trying to prove that $$ g(t) = f(x+ty) $$ is convex for all x $\epsilon $ $ S^{n}_{++}…
kann
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Let $X$ be non-empty open convex. Is $X_n := \{x \in \mathbb R^d \mid \inf_{y \in X^c} |x-y| \ge 1/n\}$ convex?

In solving this question, I have come up with the following result. Theorem: Let $X \subset \mathbb R^d$ be non-empty open convex. There is a sequence $(X_n)$ of closed convex subsets of $X$ such that $X_n \nearrow X$, i.e., $X_n \subset X$ and…
Akira
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Axiomatic approach to convex geometry

I'm looking for a text exploring/explaining an as-axiomatic/algebraic/categorical-as-possible approach to convex geometry. Everytime I open a book about convex polytopes (e.g. Ziegler, Grünbaum) I feel like I'm missing some neat axiomatic…
user76575
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Would any two consecutive extreme points on a convex hull can be linearly projected still being extreme?

Given two extreme points on a convex hull, if the straight line connects them is on the boundary , they are consecutive. Would there always exits a linear projection into lower dimension such that those two points are still extreme? For example a…
peng yu
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Intersection of 4 convex sets in $\mathbb{R}^2$

Does there exist an example of four convex sets lying in $\mathbb{R}^2$ such that the intersection of every 3 of them contains a unit-length interval but the intersection of all 4 of them does not?
blm
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Volume of polar body defined by $n$ points in $\mathbb R^n$?

Suppose I have $n$ linearly independent vectors $v_1, \ldots, v_n$ in $\mathbb R^n$. Is there an expression for the volume of the following set: $$ \{x \in \mathbb R^n : |x \cdot v_i| \leq 1 \text{ for all } i\} $$ ? My understanding is that this…
Alan C
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$S, f(S), f(f(S)),...$

For $S \subseteq \mathbb{R}^n$, let $f(S) = \{p \in \mathbb{R}^n\; |\;\text{d}(p,q) \le 1 \text{ for some }q \text{ in }S\}$. In the thread $\qquad$Given a convex set in a normed vector space, take a neighbourhood of it. Is still convex? it was…
quasi
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Non-negative map is locally uniformly continuous on the interior of a cone under hypothesis

First of all, I'm sorry if I used the wrong tags for this question, please tell me if you want me to change them. During a reading of a paper, I came across a lemma that is useful for my research (I study the characteristics of a certain function in…
NaNoS
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What is the shape of the intersection of all projections of a cube in $\mathbb{R}^3$?

Let $Q\subset \mathbb{R}^3$ denote the cube $[-1,1]\times[-1,1]\times[-1,1]$. Let $\xi\in\mathbb{S}^2$ be a vector on the unit sphere centered at the origin. Let $\xi^{\perp}$ denote the linear subspace that is orthogonal to the vector $\xi$. Denote…
VShaw
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Visualizing the Minkowski sum A + (-B)

Hello all I'm working on my own game engine, and I'm trying to wrap my mind around collision detection. I found this gem https://math.stackexchange.com/a/2203231/1098790 , but there is one thing I don't understand. It says "the Minkowski sum of 2…
gist076923
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Proof of Straszewicz's theorem

I'm interested in the following result, known as Straszewicz's theorem: (1) For $C$ a compact convex set, the set of exposed points $\text{exp}\ C$ is dense in the set of extreme points $\text{ext}\ C$. After attempting to prove this for about…
ViHdzP
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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: $$\overline{B} (0, 1) \subset T(K) \subset d \cdot \overline{B} (0, 1)?$$ The constant $d$ comes from the case of a…
D. Thomine
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