Skip to main content Accessibility help
×
Home

INTERSECTIONS OF BALLS AND SETS OF CONSTANT WIDTH IN FINITE-DIMENSIONAL NORMED SPACES

  • Horst Martini (a1), Christian Richter (a2) and Margarita Spirova (a3)

Abstract

For an arbitrary subset $X$ of a finite-dimensional real Banach space $E$ , the ball intersection with parameter $\lambda \gt 0$ is defined as the intersection of all balls of radius $\lambda $ whose centers are in $X$ . On the other hand, the intersection of all balls of radius $\lambda $ that contain $X$ is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite-dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non-empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.

Copyright

References

Hide All
1.Baronti, M. and Papini, L. P., Diameters, centers and diametrically maximal sets. Rend. Circ. Mat. Palermo (2) Suppl. 38 (1995), 1124.
2.Boltyanski, V., Martini, H. and Soltan, P. S., Excursions into Combinatorial Geometry, Springer (Berlin, 1997).
3.Boltyanski, V. and Soltan, V. P., Borsuk’s problem. Mat. Zametki 22 (1977), 621631 (in Russian).
4.Borowska, D. and Grzybowski, J., The intersection property in the family of compact convex sets. J. Convex Anal. 17 (2010), 173181.
5.Chakerian, G. D. and Groemer, H., Convex bodies of constant width. In Convexity and Its Applications (eds Gruber, P. M. and Wills, J. M.),Birhäuser (Basel, 1983), 4996.
6.Eggleston, H. G., Sets of constant width in finite dimensional Banach spaces. Israel J. Math. 3 (1965), 163172.
7.Goodey, P. R., Intersection of circles and curves of constant width. Math. Ann. 208 (1974), 4958.
8.Goodey, P. R., Intersections of planar convex curves. Math. Ann. 267 (1984), 145159.
9.Groemer, H., On complete convex bodies. Geom. Dedicata 20 (1986), 319334.
10.Grünbaum, B., Borsuk’s partition conjecture in Minkowski planes. Bull. Res. Council Israel Sect. F 7F (1957/1958), 2530.
11.Grünbaum, B., Borsuk’s problem and related questions. In Proc. Sympos. Pure Math., Vol. VII, American Mathematical Society (Providence, R.I, 1963), 271284.
12.Heil, E. and Martini, H., Special convex bodies. In Handbook of Convex Geometry, Vol. A (eds Gruber, P. M. and Wills, J. M.),North-Holland (Amsterdam, 1993), 347385.
13.Karasëv, R. N., On the characterization of generating sets. Modelir. Anal. Inf. Sist. 8 (2) (2001), 39 (in Russian).
14.Kołodziejczyk, K., Borsuk covering and planar sets with unique completion. Discrete Math. 122 (1993), 235244.
15.Lenz, H., Zur Zerlegung von Punktmengen in solche kleineren Durchmessers. Arch. Math. 6 (1955), 413416.
16.Maehara, H., Convex bodies forming pairs of constant width. J. Geom. 22 (1984), 101107.
17.Martín, P., Martini, H. and Spirova, M., Chebyshev sets and ball operators (submitted).
18.Martini, H. and Spirova, M., On the circular hull property in normed planes. Acta Math. Hungar. 125 (2009), 275285.
19.Martini, H. and Swanepoel, K. J., The geometry of Minkowski spaces—a survey, Part II. Expo. Math. 22 (2004), 93144.
20.Matoušek, J., Using the Borsuk–Ulam Theorem (Universitext), Springer (Berlin, 2003). Written in cooperation with A. Björner and G. M. Ziegler.
21.McMullen, P., Schneider, R. and Shephard, G. C., Monotypic polytopes and their intersection properties. Geom. Dedicata 3 (1974), 99129.
22.Moreno, J. P. and Schneider, R., Diametrically complete sets in Minkowski spaces. Israel J. Math. 191 (2012), 701720.
23.Moreno, J. P. and Schneider, R., Local Lipschitz continuity of the diametric completion mapping. Houston J. Math. 38 (2012), 12071223.
24.Peterson, B. B., Intersection properties of curves of constant width. Illinois J. Math. 17 (1973), 411420.
25.Raigorodskii, A. M., Three lectures on the Borsuk partition problem. In Surveys in Contemporary Mathematics (London Mathematical Society Lecture Note Series 347), Cambridge University Press (Cambridge, 2008), 202247.
26.Sallee, G. T., Sets of constant width, the spherical intersection property and circumscribed balls. Bull. Aust. Math. Soc. 33 (1986), 369371.
27.Sallee, G. T., Pairs of sets of constant relative width. J. Geom. 29 (1987), 111.
28.Spirova, M., Discrete Geometry in Normed Spaces, Südwestdeutscher Verlag für Hochschulschriften (Saarbrücken, 2011).
29.Valentine, F. A., Convex Sets, McGraw-Hill (New York, 1964).
30.Xu, C., Yuan, L. and Ding, R., Borsuk’s problem in a special normed space. Northeast Math. J. 20 (2004), 7983.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

INTERSECTIONS OF BALLS AND SETS OF CONSTANT WIDTH IN FINITE-DIMENSIONAL NORMED SPACES

  • Horst Martini (a1), Christian Richter (a2) and Margarita Spirova (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed