Benjamin Matschke

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Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler's " projecting deformed products " construction to(More)
Any continuous map of an N-dimensional simplex ∆N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ∆N to the same point in M , assuming that N ≥ (r − 1)(d + 1), no r vertices of ∆N get the same color, and our proof needs that r is a prime. A face of ∆N is called a rainbow face if all vertices have(More)
We prove that any continuous map of an N-dimensional simplex ∆N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ∆N to the same point in M : For this we have to assume that N ≥ (r − 1)(d + 1), no r vertices of ∆N get the same color, and our proof needs that r is a prime. A face of ∆N is a rainbow face if(More)
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 (trans-versal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as(More)
A short history of (tight) Tverberg theorems. The history of " Tverberg type " multiple intersection theorems (after the classical convexity results of Helly and Radon, and the non-embeddability results of van Kampen and Flores, etc.) starts with Birch's 1959 paper " On 3N points in a plane " [5], which contained the following three achievements. Theorem 1:(More)
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