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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Is circle the only shape that can remain convex after folding?

Here "fold" means "fold a piece of paper (along a straight crease)". The sketch below shows that one can always find a fold by which an ellipse or rectangle loses convexity. But it seems a circle remains convex no matter how the crease is chosen? I…
Taozi
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What are the known convex polyhedra with congruent faces?

A monohedral polyhedron is one whose faces are all congruent. Note that this is a weaker condition than being isohedral (face-transitive). We have a classification of all convex isohedral polyhedra, consisting of 30 classes of assorted finite…
RavenclawPrefect
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Which convex shapes are the hardest to bind together with a rubber band?

Suppose I have a convex set $S\subset \mathbb{R}^2$ of unit area. In fact, I have two congruent copies of $S$ which I would like to bundle together with a rubber band, i.e. take the convex hull $C$ of a disjoint union of these two copies. Assuming I…
RavenclawPrefect
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If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula rules out strictly convex $n$-gons with $n\ge 7$.…
RavenclawPrefect
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8000 congruent convex asteroids can form a "stable cluster". How much better can we do?

The Infinite Case The "Trapped in Thickland" puzzle in Peter Winkler's latest edition of Mathematical Puzzles asks the following (in my own words): The Infinite Asteroid Belt is a region in $\mathbb{R}^3$ between two parallel planes. It contains a…
greenturtle3141
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What is the difference between linearly and affinely independent vectors?

What is the difference between linearly and affinely independent vectors? Why does affine independence not imply linear independence necessarily? Can someone explain using an example?
divya garg
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Prove that $f:\mathbb R^n\to\mathbb R$ is affine if and only if it is convex and concave

Suppose $f:\mathbb{R}^n\to \mathbb{R}$ is both convex and concave, how to prove that $f$ is linear? or exactly speaking, $f$ is affine? I thought for the whole day, but I cannot figure it out. When I was working on this problem, I met another…
breezeintopl
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Frame challenge: Find the maximum $n$ such that circles of radius $1, \frac12, \frac13, ..., \frac1n$ can be held immobile by a convex frame.

Find the maximum $n$ such that circles of radius $1, \frac12, \frac13, ..., \frac1n$ can be held immobile by a convex frame, or show that there is no maximum. Here is an example with $n=7$. By "immobile", I mean no circle can move without…
Dan
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Stable strict local minimum implies local convexity

Let $\bar{x}\in\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ function. We have known that if $\nabla f(\bar{x})=0$ and $\nabla^2f(\bar{x})>0$, i.e. $\nabla^2f(\bar{x})$ is positive definite, then $\bar{x}$ is a strict local…
Blind
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What are the main ideas needed to prove that only $92$ Johnson solids exist?

The Johnson solids are the strictly convex polyhedra made out of regular faces, excluding the vertex-transitive ones (which instead enter into the uniform category). Victor Zalgaller proved in 1969, in his paper Convex Polyhedra With Regular Faces,…
ViHdzP
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$ a+b+c=0,\ a^2+b^2+c^2=1$ implies $ a^4+b^4+c^4=\frac{1}{2}$

This is a high school problem. It is solved by algebraic calculation (not difficult) - $\ast$. But I have a problem in interpretation the result. Here intersection of two surfaces $a+b+c=0,\ a^2+b^2+c^2=1$ is a circle $C$. That is, by $\ast$,…
HK Lee
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Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?

Let $V$ be a normed vector space over $\mathbb{R}$, and let $A$ and $B$ be two disjoint nonempty convex subsets of $V$. A geometric form of Hahn-Banach Theorem states that $A$ and $B$ can be separated by a closed hyperplane (i.e. there is $f \in…
Wooyoung Chin
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Find the polar set of $\{(x,y)\in\mathbb R^2: x^2+y^4\le1\}$

Let $A \subset \mathbb{R}^{n}$ be non-empty. The set $$ A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace $$ is called the polar of $A$. I'm trying to find the polar of $ \lbrace (x,y) \in…
SpicyJalapenos
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Can convexity of a polyhedron be determined solely by the line segments between its vertices?

Standard definitions state that a convex polyhedron is one where any line segment connecting two points within or on the polyhedron lies entirely within or on the polyhedron. I'm curious if this definition can be simplified when dealing specifically…
Herbert The Bird
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Inscribing Platonic solids in each other: why can't you put a dodecahedron in an octahedron?

Given convex polyhedra $P, Q$, say that one can inscribe $P$ in $Q$ if we can find points on the surface of $Q$ whose convex hull is similar to $P$. If we restrict $P, Q$ to be Platonic solids, we can achieve every case except inscribing the…
RavenclawPrefect
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