# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Thursday, October 23, 2003, 12:15 pm

**Duration**: This information is not available in the database

**Location**: This information is not available in the database

**Speaker**: Yoshio Okamoto

## Local topology of the free complex of a two-dimensional generalized convex shelling

How many interior points are there in a given d-dimensional point
configuration? For this problem, the Euler-Poincare type formula was
given by Ahrens, Gordon & MacMahon when d=2, and they conjecture it
holds for higher dimensions. This conjecture was settled
independently by Klain and by Edelman & Reiner. The proof by Edelman
& Reiner used the topological method and they raised open problems for
the further generalization to the so-called "convex geometries" which
were first introduced by Edelman & Jamison.

In this work, we consider
a subclass of the convex geometries which consists of the
2-dimensional separable generalized convex shelling, and show the
conjecture is valid for this class. This is not just a special case
but a step toward a resolution of their open problems because it was
shown by Kashiwabara, Nakamura & Okamoto that every convex geometry is
isomorphic to some d-dimensional separable generalized convex shelling
for some d, which we call the representation theorem of convex
geometries.

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