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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
.
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
. As already Rogers
observed in 1967, this result implies that
for
centrally symmetric convex bodies. Similarly,
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
of Januszewski [1998]
proving
for centrally symmetric convex bodies,
and
for arbitrary d-dimensional convex
bodies.
Next: Anke Walz - The Up: Discrete Geometry / Géométrie Previous: Peter Schmitt - The
