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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)

T his is a short survey article on a 103-year-old and still open problem in plane geometry, the Square Peg Problem. It is also known as the Inscribed Square Problem and it is due to Otto Toeplitz. Every continuous simple closed curve in the plane γ : S 1 → R 2 contains four points that are the vertices of a square. A continuous simple closed curve in the… (More)

- Pavle V M Blagojevi´c, Benjamin Matschke, Günter M Ziegler, Résumé
- 2011

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)

- Pavle V M Blagojevi´c, Benjamin Matschke, Günter M Ziegler
- 2010

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)

- Günter M Ziegler, Pavle V M Blagojevi´c, Benjamin Matschke, Florian Frick
- 2014

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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