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A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

2 votes

Convex bodies have more volume on the outside near the boundary

A modification of Victor Protsak's answer yields a short and almost elementary proof: Since the set of convex $n$-dimensional convex polytopes is dense in the space of all $n$-dimensional convex bodie …
Wlodek Kuperberg's user avatar
4 votes

Is a polytope that has in-spheres for faces of all dimensions already regular?

In $R^3$, since the spheres are concentric, not only all faces are regular, but also all edges are of the same length, and all faces are inscribed in circles of the same radius, hence are congruent. A …
Wlodek Kuperberg's user avatar
2 votes
Accepted

Closed form solutions for maximal subsets of convex polytopes

By Ivan's request, I describe here two algorithms for finding the maximum-size ball contained in a given $n$-dimensional convex polytope $P$. The first algorithm, requiring central symmetry of $P$, pr …
Wlodek Kuperberg's user avatar
17 votes

How to make a sandwich from just one piece of bread?

This won't be a complete answer to Q2, but something of a starting point, at least for all two-dimensional convex $P$. Assuming for simplicity that the area of $P$ is 1, $P$ contains a parallelogram o …
Wlodek Kuperberg's user avatar
3 votes
Accepted

On necessary condition for no integer points in polytope

The answer is $no$. In the plane, the circle of diameter $d$ greater than $1$ but smaller than $\sqrt2$ and centered at $({1\over2},{1\over2})$ contains no point with integer coordinates, and the squ …
Wlodek Kuperberg's user avatar
11 votes

Triangle with largest perimeter in a convex region

$\bullet$ Every convex region of area $1$ contains a triangle of area at least as large as the area of the equilateral triangle inscribed in the circle of area $1$. Moreover, if the region does not co …
Wlodek Kuperberg's user avatar
3 votes
Accepted

Map from a convex polygon that increases distance

Begin with a lemma: Let $\ T=\Delta oab$ be a triangle contained in a circle centered at the origin $o$ and let the arc $\alpha$ of the circle be such that the circle's sector $S$ based on $\alpha$ h …
Wlodek Kuperberg's user avatar
2 votes

Maximizing ratio volume/diameter^n by an affinity

A closely related problem, considering the $(n-1)$-dimensional surface area instead of the diameter of the body, has been solved by Keith Ball, Volume ratios and a reverse isoperimetric inequality. J. …
Wlodek Kuperberg's user avatar
10 votes

A convex curve inside the unit circle

The drawing below illustrates a proof for the theorem as stated in your question; almost no words of explanation are necessary. To make things easy, I write a few lines. The whole proof is based on th …
Wlodek Kuperberg's user avatar
7 votes

Two cubes in unit cube

A better bound, namely $\ \ \displaystyle a+b\le\frac{2\sqrt n}{\sqrt n+1}\ \ $ is obtained as follows. The cubes of edge lengths $a$ and $b$ contain inscribed balls of diameters $a$ and $b$, respec …
Wlodek Kuperberg's user avatar