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Benjamin Matschke

  1. On the Square Peg Problem and some Relatives.

    Втр, 01/05/2010 - 16:01 — SomNambul
    Authors: Benjamin Matschke
    Subjects: Metric Geometry
    Abstract

    The Square Peg Problem asks whether every continuous simple closed planar
    curve contains the four vertices of a square. This paper proves this for the
    largest so far known class of curves.

  2. A Note on Masspartitions by Hyperplanes.

    Втр, 01/05/2010 - 16:01 — SomNambul
    Authors: Benjamin Matschke
    Subjects: Combinatorics
    Abstract

    A triple of positive integers (d,h,m) is admissible if for any m given masses
    in R^d there exist h hyperplanes that cut each of these masses into 2^h equal
    pieces. We present an elementary reduction which combined with results by Ramos
    (1996) yields all the admissible triples that were known up to now (with one
    exception) as well as new ones.

  3. Optimal bounds for a colorful Tverberg--Vrecica type problem.

    Пнд, 11/16/2009 - 16:01 — SomNambul
    Authors: Benjamin Matschke, Pavle Blagojevic, Gunter Ziegler
    Subjects: Algebraic Topology
    Abstract

    We prove the following optimal colorful Tverberg--Vrecica type transversal
    theorem. For a prime $r$ and $k+1$ colored collections $S_i=\biguplus S_i^j$,
    $|S_i|=(r-1)(d-k+1)$, $|S_i^j|\leq r-1$, $i=0,...,k$, of points in $\R^d$ there
    exist a partition of each collection $S_i$ into colorful sets
    $T_i^1,...,T_i^{r}$ with a $k$-plane meeting all their convex hulls
    $\conv(T_i^j)$, under the assumption that $r(d-k)$ is even or $k=0$.

  4. Prodsimplicial-Neighborly Polytopes.

    Пнд, 08/31/2009 - 17:16 — SomNambul
    Authors: Benjamin Matschke, Julian Pfeifle, Vincent Pilaud
    Subjects: Metric Geometry
    Abstract

    We introduce PSN polytopes, whose k-skeleton is combinatorially equivalent to
    that of a product of r simplices. They simultaneously generalize both
    neighborly and neighborly cubical polytopes.

    We construct PSN polytopes by three different methods, the most versatile of
    which is an extension of Sanyal and Ziegler's "projecting deformed products"
    construction to products of arbitrary simple polytopes. For general r and k,
    the lowest dimension we achieve is 2k+r+1.

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