Seminar

In this talk I will present results that were obtained and posted during my stay at MSRI in Fall of 2017 in the program Geometric and Topological Combinatorics.

The main part of the talk will be devoted to the results with Nevena Palic and Guenter Ziegler on Cutting a part from many measures, arXiv:1710.05118: 

Holmsen, Kyncl and Valculescu recently conjectured that if a subset X  of the d-dimensional Euclidean space E with kn points colored by m different colors can be partitioned into n subsets of k points each, such that each subset contains points of at least d different colors, then there exists such a partition of X with the additional property that the convex hulls of the n subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least c different colors, where we also allow c to be greater than d. Indeed, for integers m,c and d and a prime power n=p^k such that m >\ n(c-d)+dn/p-n/p+2, and for m given positive finite absolutely continuous measures on E, we prove that there exists a partition of E into n convex sets, such that every set has positive measure with respect to at least c of the measures.

The second part of the talk will be used to illustrate results with Djordje Baralic, Roman Karasev and Aleksandar Vucic on Orthogonal shadows and index of Grassmann manifolds, arXiv:1709.10492:

We study the Z/2 action on real Grassmann manifolds of n-planes in (2n)-dimensional Euclidean space E given by taking orthogonal complement. We completely evaluate the related Z/2 Fadell--Husseini index utilizing a novel computation of the Stiefel--Whitney classes of the wreath product of a vector bundle.

These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For n=2^a (2b+1), k=2^{a+1}-1, C a convex body in E, and k real valued functions a_1,…,a_k continuous on convex bodies in E with respect to the Hausdorff metric, there exists an n-dimensional subspace V of E such that projections of C to V and its orthogonal complement V’ have the same value with respect to each function a_i.