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Istvan Talata - On translative coverings of a convex body with its homothetic copies of given total volume



ISTVAN TALATA, Department of Mathematics, Auburn University, Auburn, Alabama  36849-5310, USA
On translative coverings of a convex body with its homothetic copies of given total volume


Let K be a d-dimensional convex body. Denote by h(K) the minimum number of smaller homothetic copies of K which are needed to cover K. Furthermore, denote by hv(K) the smallest real number with the property that every sequence of positive homothetic copies of K with total volume at least hv(K) vol(K) permints a translative covering of K. It is clear that $h(K)\leq hv(K)+1$.

It is proved by Rogers [1957] that for any d-dimensional convex body K there exists a covering of Rd with translates of K with density at most $\delta_d=d\log d+d\log \log d+5d$. As already Rogers observed in 1967, this result implies that $h(K)\leq 2^d\delta_d$ for centrally symmetric convex bodies. Similarly, $h(K)\leq 4^d\delta_d$was proved for arbitrary convex bodies.

In this talk we show that the method used by Rogers can be extended for homothetic copies of K with different coefficients. This way we can improve on the upper bound $hv(K)\leq (d+1)^d-1$ of Januszewski [1998] proving $hv(K)\leq 2^{d+o(d)}$ for centrally symmetric convex bodies, and $hv(K)\leq 4^{d+o(d)}$ for arbitrary d-dimensional convex bodies.


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Next: Anke Walz - The Up: Discrete Geometry / Géométrie Previous: Peter Schmitt - The
 

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