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)
A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane or, equivalently, a topological embedding S1 ↩ R2. In its full generality Toeplitz’s problem is still open. So far it has been solved affirmatively for curves that are “smooth enough” by various authors for(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 note that Birch’s Theorem 1*, as well as his conjecture, which was proved in full by Helge Tverberg in 1964, fifty years ago (see [17]), and thus is now known as Tverberg’s theorem [13], are tight : This is not only evident from concrete configurations, but also from a general position argument: If (r− 1)(d+ 1) points in R in general position are(More)
Erdős-Szekeres theorem is one of classic results in combinatorial geometry. In this project we consider the colored version of the problem. Especially, we are interested in the number of empty monochromatic triangles and the existence of an empty monochromatic convex quadrilateral. We give some minor results and plausible ideas to solve the problems.
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 different(More)
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