Benjamin Matschke

The width of five-dimensional prismatoids

Santos’ construction of counter-examples to the Hirsch conjecture is based on the existence of prismatoids of dimension d of width greater than d. The case d = 5 being the smallest one in which this can possibly occur, we here study the width of 5-dimensional prismatoids, obtaining the following results: • There are 5-prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos’ original construction. This leads to lowering the dimension of the non-Hirsch polytopes from 43 to only 20. • There are 5-prismatoids with n vertices and width ( p n) for arbitrarily large n.

A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

We present a parametrized version of Volovikovs powerful Borsuk–Ulam–Bourgin–Yang type theorem, based on a new Fadell–Husseini type ideal-valued index of G-bundles which makes computations easy. As an application we provide a parametrized version of the following waist of the sphere theorem due to Gromov, Memarian, and Karasev–Volovikov: Any map f from an n-sphere to a k-manifold (n ≥ k) has a preimage f-1(z) whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator Sn-k (if n = k we further need that f has even degree).

Projective Center Point and Tverberg Theorems

We present projective versions of the center point theorem and Tverberg’s theorem, interpolating between the original and the so-called “dual” center point and Tverberg theorems. Furthermore we give a common generalization of these and many other known (transversal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as some essentially new results about partitioning measures in projective space.