# Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, July 27, 2004, 12:15 pm

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

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

**Speaker**: Volker Kaibel (TU Berlin)

## Low-dimensional faces of random 0/1-polytopes

Let *P* be a random 0/1-polytope in *R*^{d} with *n(d)* vertices, and
denote by *φ*_{k}(P) the *k*-face density of *P*, i.e.,
the quotient of the number of *k*-dimensional faces of *P* and
*\binom{n(d)}{k+1}*. For each *k >= 2*, we establish the existence
of a sharp threshold for the *k*-face density and determine the
values of the threshold numbers *τ*_{k} such that, for all
*ε > 0*,

E(*φ*_{k}(P)) =

- 1- o(1), if
*n(d)<= 2*^{(τk-ε)d} for all *d*
- o(1), if
*n(d)>= 2*^{(τk+ε)d} for all *d*

holds for the expected value of

*φ*_{k}(P). The threshold for

*k=1* has recently been determined by K. and Remshagen (2003).

In particular, these results indicate that the high face densities
often encountered in polyhedral combinatorics (e.g., for the
cut-polytopes of complete graphs) should be considered more as a
phenomenon of the general geometry of 0/1-polytopes than as a
feature of the special combinatorics of the underlying problems.

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