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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.

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.

1433 questions
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Convex side of a spherical mirror

A convex set has the property that if you take any two points in the set and draw the line segment connecting those two points, that line segment lies entirely in the set. My textbook says that the figure on top represents a convex mirror and…
Siddhartha
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Intersection of two convex sets

Let $A$ and $B$ be disjoint finite subsets of $\mathbb{R}^d$ for some $d$. Furthermore, assume that $|A\cup B|\geq d+3$, $conv(A)\cap conv(B)\neq\emptyset$, and points in $A\cup B$ are in general position, i.e., any $d+1$ points of them are affinely…
MathFun
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Is the logarithm of Banach-Mazur distance between convex bodies an actual distance?

Let $K,L\in\mathcal{K}^n$ be two convex bodies, their Banach-Mazur distance is defined as $$d_{BM}(K,L)=\inf\{\lambda>0\colon\exists x,y\in\mathbb{R}^n,T\in GL_n(\mathbb{R})/y+K\subset T(x+L)\subset\lambda(y+K)\}.$$ Where $GL_n(\mathbb{R}^n)$ is the…
F.Webber
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Generalization of Sylvester–Gallai theorem

According to the Sylvester–Gallai theorem, given a finite number of points in the Euclidean plane, either: 1) all the points are collinear; or \ 2)there is a line which contains exactly two of the points. \ Now, I want to know, is it possible to…
123...
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If a point in a set can be element-wisely smaller than a point and larger than other point, then whether the point is an extreme point or not?

As the title, I think, in a closed, bounded, and convex set $C$ in $\mathbb{R}^n$, $C=\{\sum_{i=1}^{k}\lambda_k x_k:x_i\in C_i, \lambda_i\geq 0, \sum_{i=1}^{k}\lambda_k\leq 1\}$, where $C_i$ is a convex, bounded, and closed set for all…
jerry
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What is the radius of the largest inscribed disc of a given projection of a cube in $\mathbb{R}^3$?

Let $Q$ be a cube in $\mathbb{R}^3$ centered at the origin with side length $l$. Let $\psi\in\text{SO}(3)$ so that $\psi(Q)$ is a rotation of $Q$ in the space $\mathbb{R}^3$. Denote $\pi(\psi(Q))$ as the projection of the rotated cube onto the…
VShaw
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How to show that you can always move away from a convex object?

Assume I have two convex objects in 3D, $A$ and $B$, and the two are in contact. How can I show that there always exists a vector $v$, such that if I move $A$ along $v$ the contact between $A$ and $B$ is broken?
QualsPassed
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Best constants in the Sudakov inequalities

The Sudakov inequalities bound the number of scaled Euclidean balls needed to cover a centrally symmetric convex body $K \subset \mathbb{R}^n$, and vice versa. Before I state them, let me say what I'm interested in: the best possible constant $c$ in…
Spencer Peters
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Circle enclosing the convex hull of $n$ points

Suppose $P_1,P_2,,\cdots,P_n$ aren points inn the plane with given centriod $C=(a,b)$and $$d=\text{max distance}(P_i,P_j),i \neq j$$.Can we find a circle of some radius and center in terms of $d$ and $a$ and $b$ which contains the convex hull of…
AgnostMystic
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How to prove an inequality of gauge norms?

Let $A$ be a compact convex set in $\mathbb R^n$. Let $y\in \mathbb R^n$ be an arbitrary point not belonging to $A$. Let $P$ be a hyper-plane which separates $A$ and $y$. Let $x$ be the projection of $y$ onto $P$. We now recall the definition of…
Leonard Neon
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determine if high dimensional object is a Zonotope via low dimensional projection.

Supposedly I have a Convex object $A$ in high dimension(Eculdian space), and I know their low dimensional projection(only look at 2 dimensions at a time) is a zonotope, how can I assert/reject $A$ is a Zonotope? A concrete example, $$ A = CH(\{(0,…
peng yu
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Interior of Minkowski sum of closed convex sets

I'd like to proof the following statement: Let $A, B \subset \mathbb{R}^n$ both be convex and closed. Additionally, let $A$ be full-dimensional (e.g. $\text{int}(A) \neq \emptyset$). Then $\text{int}(A+B) = \text{int}(A) + B$. The inclusion…
Markus Müller
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How to prove that in a closed, convex, and not bounded set from every point there is a closed half-line that is in the set?

Let $K \subseteq \mathbb{E}^n$ a closed, convex, and not bounded set. How to show that then from every point of $K$ there is a closed half-line that is in $K$.
cizitebev
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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 $K+\mathrm {cl}K=K?$
Madara Uchiha
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The sum over inverses of n positive real numbers, which is less or equal 1, yields a convex set

Is the set $\left\{(x_1,x_2,\dots,x_n)\in\mathbb{R}_{\geq 1}^n:\sum\limits_{i=1}^n\frac{1}{x_i}\leq 1\right\}$ convex? For $n=2$ it is not too difficult to show convexity and I guess one has to find the right transformations to avoid messy…
Jan
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