On a covering property of convex sets

Groemer, H.

doi:10.1090/S0002-9939-1976-0412970-2

On a covering property of convex sets
HTML articles powered by AMS MathViewer

by H. Groemer PDF

Proc. Amer. Math. Soc. 59 (1976), 346-352 Request permission

Abstract:

Let $\{ {K_1},\;{K_2}, \ldots \}$ be a class of compact convex subsets of euclidean $n$-space with the property that the set of their diameters is bounded. It is shown that the sets ${K_i}$ can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets ${K_i}$ is infinite. Also, some statements regarding the densities of such coverings are proved.

References

Gulbank D. Chakerian, Covering space with convex bodies, The geometry of metric and linear spaces (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1974) Lecture Notes in Math., Vol. 490, Springer, Berlin, 1975, pp. 187–193. MR 0407722
G. D. Chakerian and H. Groemer, On classes of convex sets that permit plane coverings, Israel J. Math. 19 (1974), 305–312. MR 367818, DOI 10.1007/BF02757494
Helmut Groemer, Existenzsätze für Lagerungen im Euklidischen Raum, Math. Z. 81 (1963), 260–278 (German). MR 163222, DOI 10.1007/BF01111546
H. Hadwiger, Volumschätzung für die einen Eikörper überdeckenden und unterdeckenden Parallelotope, Elem. Math. 10 (1955), 122–124 (German). MR 77154
A. Kosiński, A proof of an Auerbach-Banach-Mazur-Ulam theorem on convex bodies, Colloq. Math. 4 (1957), 216–218. MR 86324, DOI 10.4064/cm-4-2-216-218