9
votes
3answers
697 views

Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably also define a special class of Pythagorean triples. A perfect squared square PSS is a square (as a plane figure) partitioned into smaller ...
6
votes
1answer
95 views

Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
10
votes
1answer
258 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
12
votes
0answers
120 views

realization spaces of 3-dimensional polytopes

It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible. A proof of this theorem can be found for instance in ...
7
votes
1answer
439 views

Separating pairs of points in R^n

Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r_n>0$ one ...
6
votes
0answers
178 views

Right-angled polytopes

%This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ...
12
votes
2answers
399 views

Sets of evenly distributed points in the Euclidean plane

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite? If the answer is yes, can ...
3
votes
0answers
92 views

Generalized flag complex?

Assume we glue an $n$-dimensional simplicial complex $K$ from copies of an $n$-simplex $\Delta$ with fixed spherical metric. We may think that $\Delta$ has colored vertices and we glue so that the ...
3
votes
2answers
227 views

Problems similar to Borsuk’s Theorem in the plane

Consider a 2-dimensional Borsuk's theorem: Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$. I wonder if there are any ...
10
votes
1answer
254 views

The number of relevant scales for a finite metric space

For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. ...
1
vote
1answer
156 views

Pythagorean triples related to non-isometric equidistant plane quadruples

QUESTION   Do there exist integers   $u\ x\ A\ B$   such that   $x\ne 0$,   and the following two equalities hold: $ x^2 + (x-u)^2\ =\ A^2$ $ x^2 + (x+u)^2\ =\ B^2$ ? ...
9
votes
1answer
251 views

Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
4
votes
1answer
213 views

A regular polytope

For positive integers $m$ and $n$, consider a regular polytope in ${\mathbb R}^{m+n+mn}$ with $2^{m+n}$ vertices, corresponding to each $\sigma \in \{-1,1\}^{m+n}$ as follows. The first $m+n$ ...
7
votes
0answers
122 views

Minimal spanning tree of a point set in the unit square, under an unusual distance function

For two points $x$, $y \in [0,1]^2$, let their distance be $d(x,y) := \|x-y\|_2^2$ (i.e. the usual distance, squared). Technically, this is a semimetric, as it does not satisfy the triangle ...
5
votes
2answers
242 views

Kissing Number of Spheres in Non-Euclidean Geometry

There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ ...
1
vote
1answer
418 views

Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is $$\inf_{\begin{array}{c} v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\ v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right) \end{array}}\max_{0\le ...
7
votes
0answers
270 views

Volume-like property to upper bound lattice points in a convex body

The following question arises in passing in a joint paper that I am working on. Let $K$ be a centrally symmetric convex body in an $n$-dimensional real vector space $V$ which contains a lattice $L$. ...
2
votes
2answers
224 views

What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of subsets in a subdivision of the set of $30$ points into subsets such that all the points in each subset are on the boundary of the ...
11
votes
1answer
329 views

Can all convex polytopes be realized with vertices on surface of convex body?

The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited ...
1
vote
0answers
141 views

dissections and vertices of non-convex polytopes

Let us call a finite union $P$ of $n$-dimensional compact convex polytopes in $\mathbb{R}^n$ a non-convex polytope. Recall that a dissection of $P$ is a finite collection $T$ of $n$-dimensional ...
20
votes
1answer
1k views

A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter? Background and motivation The Borsuk conjecture (disproved in ...
7
votes
4answers
523 views

Tessellating $\mathbb{R}^n$ by bricks.

Let us call the $\ell_1$-product of intervals $[0,k_1]\times...\times [0,k_n]$ a brick of size $k_1+...+k_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point ...
6
votes
1answer
428 views

Doubling dimension of a Euclidean space

The doubling dimension of a metric space $X$ is the smallest positive integer $k$ such that every ball of $X$ can be covered by $2^k$ balls of half the radius. It is well known that the doubling ...
17
votes
4answers
656 views

The Constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
14
votes
2answers
576 views

Random permutations from Brownian motion

Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation ...
3
votes
3answers
236 views

measuring n 2-planes in R^{2n}.

Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...
9
votes
3answers
647 views

Point sets in Euclidean space with a small number of distinct distances

It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
12
votes
1answer
579 views

Applications of the GCD metric

In the pre-MO era, I once realized that on the integers, the function $$ d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ , $$ is a metric (all properties are easily verified; in fact ...
4
votes
0answers
402 views

Intersection of pencils in $\mathcal{R}^2$

Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...
7
votes
2answers
292 views

non-rigidity of interior points in polyhedral triangulations?

It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into ...
4
votes
1answer
394 views

When can a 3-dimensional triangulation be isometricaly embedded in R^n?

Consider a triangulation of some bounded region of $R^3$ with a (finite) set of tetrahedra (like in Regge calculus). It can be thought of as a simplicial 3-complex with specified lengths of edges. The ...
1
vote
1answer
273 views

Large subgroups of the Hamming cube

Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube). Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
7
votes
1answer
350 views

The volume of the “unit ball” in $\mathbb{R}^{m\times n}$ with respect to the cut norm

This question is inspired by the question “ε-nets with respect to the cut norm” by the user Aaron, which had been reposted to cstheory.stackexchange.com. The cut norm ||A||C of a matrix A=(aij)∈ℝm×n ...
13
votes
0answers
457 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in ...
3
votes
0answers
191 views

How many set partitions on a big cube’s boundary arise from cubomino decompositions of the solid cube?

Introduction. This is a counting question about configurations that can appear on the outside of assembled Soma cube-like puzzles. More specifically, it’s about the ways in which the pieces of an ...
14
votes
3answers
1k views

Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.

Is the following fact true? Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k ...
2
votes
4answers
184 views

How to compare finite point sets in normed spaces?

I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define $$ d(A, B) := ...
2
votes
1answer
2k views

Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]

How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
5
votes
2answers
502 views

Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.

Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries? The answer is easy for ...
11
votes
3answers
746 views

To what extent is convexity a local property?

A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
20
votes
2answers
5k views

Is the Jaccard distance a distance?

Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...
5
votes
3answers
588 views

How can I embed an N-points metric space to a hypercube with low distortion?

I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
12
votes
3answers
975 views

distance regular metric spaces

A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
9
votes
1answer
508 views

Which changes of metric fix all open balls of a metric space?

In an earlier question, I was interested in counting the number of metric spaces on N points, where I considered two metric spaces to be the same if they had the same collection of open balls. Two ...