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

Date and Time: Thursday, December 04, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Yves Brise

Covering Spheres with Spheres

We consider $\vartheta$, the minimum density of a covering of some $d$-sphere by unit balls in $\mathbb{R}^{d+1}$. Intuitively speaking the covering density is the average number of balls containing any given point on the sphere. In an older result, around 1960, Rogers has shown that $\vartheta$ is at most $d \ln d$ plus some lower order terms. This upper bound has just recently been improved, by Dumer in 2007, to $1/2 d \ln d$ plus lower order terms. This is only an improvement in the constant of the leading term, but still remarkable for such a long standing upper bound as the Rogers bound. The lower bound $\vartheta \in \Omega(d)$ is from 1959 by Coxeter, Few, and Rogers. If time permits, we will also speak about this.

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