What does a typical metric space look like?

The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the metric polytope for large $n$ and show that it is close to the cube $[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$ in the following two senses. First, the volume of the polytope is not much larger than that of the cube, with the following quantitative estimates: $$\left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}).$$ Second, when sampling a metric space from $\mathcal{M}_n$ uniformly at random, the minimum distance is at least $1 - n^{-c}$ with high probability, for some $c > 0$. Our proof is based on entropy techniques. We discuss alternative approaches to estimating the volume of $\mathcal{M}_n$ using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.Comment: 64 pages, 2 figures. v2: Swapped Sections 5 and 6 and added a reader's guid

Slices, slabs, and sections of the unit hypercube

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference

Automated Design of Tissue Engineering Scaffolds by Advanced CAD

The design of scaffolds with an intricate and controlled internal structure represents a challenge for Tissue Engineering. Several scaffold manufacturing techniques allow the creation of complex and random architectures, but have little or no control over geometrical parameters such as pore size, shape and interconnectivity- things that are essential for tissue regeneration. The combined use of CAD software and layer manufacturing techniques allow a high degree of control over those parameters, resulting in reproducible geometrical architectures. However, the design of the complex and intricate network of channels that are required in conventional CAD, is extremely time consuming: manually setting thousands of different geometrical parameters may require several days in which to design the individual scaffold structures. This research proposes an automated design methodology in order to overcome those limitations. The combined use of Object Oriented Programming and advanced CAD software, allows the rapid generation of thousands of different geometrical elements. Each has a different set of parameters that can be changed by the software, either randomly or according to a given mathematical formula, so that they match the different distribution of geometrical elements such as pore size and pore interconnectivity. This work describes a methodology that has been used to design five cubic scaffolds with pore size ranging from about 200 to 800 µm, each with an increased complexity of the internal geometry.Mechanical Engineerin

A stability result for the cube edge isoperimetric inequality

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon$-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.Comment: 12 page

Integer cells in convex sets

Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Z^n. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.Comment: Historical remarks on the notion of the combinatorial dimension are added. This is a published version in Advances in Mathematic