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13 votes
0 answers
751 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 J}a_{i,j}\...
Aaron's user avatar
  • 794
6 votes
0 answers
1k views

How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?

Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
Wolfgang's user avatar
  • 13.4k
5 votes
0 answers
313 views

Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes

Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement? There exists ...
Penelope Benenati's user avatar
5 votes
0 answers
199 views

Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size

Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$. Let $s(C)$ and $\ell(C)$ be ...
Penelope Benenati's user avatar
4 votes
0 answers
144 views

Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$

We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
Penelope Benenati's user avatar
3 votes
0 answers
134 views

Two questions on counterexamples to Borsuk's conjecture and ball-packings

In 1933 Karol Borsuk conjectured the following Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$? Whilst new to this ...
Felix's user avatar
  • 31
2 votes
0 answers
48 views

Maximum coverage of an orthogonal polygon using $k$ rectangles

I have an orthogonal polygon (all edges are horizontal or vertical) which is convex (no holes in any row of column of the polygon). I would like to cover as much as possible of this orthogonal polygon ...
user536106's user avatar
2 votes
0 answers
131 views

Optimal way to group points in the plane into clusters

Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
Mohan Swaminathan's user avatar