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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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 that we can pick the separating hyperplane in this…
chhro
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Is this a convex set?

Is set A convex when set A = the union of all points on a circle and all points in the interior of the circle, minus point R?
Jolly
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A problem regarding a convex closed set

When studying about convex sets, I encountered this problem: Given a convex closed set $C \subset \mathbb{R}^n$. Prove that there exists a family of hyperplanes $\{ H_ i \}$ ($i \in I$) with $$H_i = \{ x \in \mathbb{R}^n, \langle a_i,x \rangle =…
ElementX
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Geometric proof of Caratheodory's theorem

I read wikipedia's proof of Caratheodry's theorem for convex sets. I was wondering if there exists a more geometric proof. I was thinking of something along the following line of reasoning. The theorem: Given a set $P$ of points in $R^d$. A point…
Jeff
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On the volume of sections of symmetrical convex bodies

The problem. Suppose you have a centrally symmetrical (with respect to the origin) convex body in $\mathbb{R}^n$, and you take the sections by intersecting it with hyperplanes in a fixed direction $u$. I want to prove that the section with the…
F.Webber
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Find a star convex set that is not convex.

Find a star convex set that is not convex. I already saw the answer on the internet but it is not clear to me, the answer is simply a figure: I would like to know why this set is the one that serves, my intuition tells me that it is because the…
user424241
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Quasi-concavity of a function

I'm trying to show the following function is quasi-concave in $x$ $$\sum^n_{i=0}{n\choose i}F(x)^{i}(1-F(x))^{n-i}u(i)$$ Here, $x$ is defined on $[0,1]$ and $F$ maps $[0,1]$ to $[0,1]$, and is a strictly increasing in $x$. So, the function is a…
Andeanlll
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Intersection of a closed convex set and an affine set

Is the intersection of a closed convex set of $\mathbb{R^n}$ and an affine set $\mathbb{R^k} \subset \mathbb{R^n}, k \le n$ a closed convex set of $\mathbb{R^n}$? (closed convex sets are convex sets that contain all their limit points.)
user451674
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how to figure out plane convex domain if all length of its intersection with coordinate lines is known?

Suppose $D\subset[0,1]\times[0,1]$ is a plane convex domain, define two function on $[0,1]$ related to $D$ as $$ h(x) = \text{length of line segment } D \cap \{ (x,y) | 0\leq y\leq 1 \} $$ $$ w(y) = \text{length of line segment } D \cap \{ (x,y) |…
wagermn
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Which of the following functions is quasiconvex?

a) $F(x)=x^{2}$ b) $F(x)=e^{-x}$ c) $F(x)=\cos (x)$ d) $F(x)=x^{-1}$ if $f\neq 0$ and $f=0$ if $x=0$ Hi, i am finding quasi concavity and convexity very difficult to understand. Any help with this will be appreciated.
Idkwoman
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About support function of convex bodies.

My question is about the support function defined $$s_{A}(x)= \sup \{x\cdot a | a \in A\}$$ ($A\subset R^{n}, x \in S^{n}$).(more generally it can be taken to be other scalar product.) So the function is defined on sets, and generally for the convex…
kolobokish
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