Covering the boundary of a convex set by tiles
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- by Marek Lassak
- Proc. Amer. Math. Soc. 104 (1988), 269-272
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958081-7
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Abstract:
Let Euclidean space ${E^n}$ be dissected by a finite family of hyper-planes. We estimate how many of the tiles obtained are sufficient to cover the boundary of a convex set $A \subset {E^n}$.References
- G. L. Alexanderson and John E. Wetzel, Divisions of space by parallels, Trans. Amer. Math. Soc. 291 (1985), no. 1, 363–377. MR 797066, DOI 10.1090/S0002-9947-1985-0797066-2
- V. G. Boltjansky and I. Ts. Gohberg, Results and problems in combinatorial geometry, Cambridge University Press, Cambridge, 1985. Translated from the Russian. MR 821465, DOI 10.1017/CBO9780511569258
- G. D. Chakerian and S. K. Stein, Some intersection properties of convex bodies, Proc. Amer. Math. Soc. 18 (1967), 109–112. MR 206818, DOI 10.1090/S0002-9939-1967-0206818-3 B. Grünbaum, Borsuk’s problem and related questions, Convexity, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 271-284. H. Hadwiger, Ungeloste Probleme. Nr 20, Elem. Math. 12 (1957), 121.
- Marek Lassak, Solution of Hadwiger’s covering problem for centrally symmetric convex bodies in $E^3$, J. London Math. Soc. (2) 30 (1984), no. 3, 501–511. MR 810959, DOI 10.1112/jlms/s2-30.3.501
- F. W. Levi, Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns, Arch. Math. (Basel) 6 (1955), 369–370 (German). MR 76368, DOI 10.1007/BF01900507
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 J. Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math. 1 (1826), 349-364.
- Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 357135, DOI 10.1090/memo/0154
- Thomas Zaslavsky, A combinatorial analysis of topological dissections, Advances in Math. 25 (1977), no. 3, 267–285. MR 446994, DOI 10.1016/0001-8708(77)90076-7
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 269-272
- MSC: Primary 52A45; Secondary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958081-7
- MathSciNet review: 958081