The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls)

Limitations to Frechet's Metric Embedding Method

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur

Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen--type obstructions) as developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks adde

Lower Bounds for L1L_1 Discrepancy

We find the best asymptotic lower bounds for the coefficient of the leading term of the $L_1$ norm of the two-dimensional (axis-parallel) discrepancy that can be obtained by K.Roth's orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, complex and harmonic analysis.Comment: a slightly different version of the article is accepted to "Mathematika

Combinatorial Stokes formulas via minimal resolutions

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn implies "Dold's theorem" that there is no equivariant map from an n-connected to an n-dimensional free Z_k-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).Comment: 18 page