On Constant Width Sets in Hilbert Spaces and Around

Luigi Caspani; Pier Luigi Papini

doi:10.68381/jca22046

Abstract

We present some results concerning the family

\mathcal{CW}

of sets with constant width and the family

\mathcal{DM}

of diametrically maximal (or diametrically complete) sets. In particular we prove (Theorem 2) that

\mathcal{CW} = \mathcal{DM}

in Hilbert spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Luigi Caspani

Dip. di Matematica "F. Enriques", Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy

luigi.caspani@gmail.com

Pier Luigi Papini

Dip. di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

pierluigi.papini@unibo.it

Keywords

Constant width
diametrically maximal sets
arcs

Mathematics Subject Classification

52A05; 52A21, 46B99

References

G. Averkov: On cross-section measures in Minkowski spaces, Extr. Math. 18 (2003) 201–208.Google Scholar
M. Baronti, P. L. Papini: Diameters, centers and diametrically maximal sets, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 38 (1995) 11–24.Google Scholar
G. D. Chakerian, H. Groemer: Convex bodies of constant width, in: Convexity and its Applications, P. M. Gruber, J. M Wills (eds.), Birkhäuser, Basel (1983) 49–96.https://doi.org/10.1007/978-3-0348-5858-8_3 Google Scholar
L. Dalla, N. K. Tamvakis: Sets of constant width and diametrically complete sets in normed spaces, Bull. Greek Math. Soc. 26 (1985) 27–39.Google Scholar
C. Franchetti: Relationship between the Jung constant and a certain projection constant in Banach spaces, Ann. Univ. Ferrara, N. Ser., Sez. VII 33 (1977) 39–44.https://doi.org/10.1007/BF02825987 Google Scholar
P. Gritzmannn, V. Klee: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces, Discrete Comput. Geom. 7 (1992) 255–280.https://doi.org/10.1007/BF02187841 Google Scholar
A. S. Granero, J. P. Moreno, R. R. Phelps: Mazur sets in normed spaces, Discrete Comput. Geom. 31 (2004) 411–420.https://doi.org/10.1007/s00454-003-0808-5 Google Scholar
J. P. Moreno, P. L. Papini, R. R. Phelps: Diametrically maximal and constant width sets in Banach spaces, Can. J. Math. 58 (2006) 820–842.https://doi.org/10.4153/CJM-2006-033-1 Google Scholar
J. P. Moreno, P. L. Papini, R. R. Phelps: New families of convex sets related to diametral maximality, J. Convex Analysis 13 (2006) 823–837.https://doi.org/10.68381/jca13061 Google Scholar
H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces - a survey. Part II, Expo. Math. 22 (2004) 93–144.https://doi.org/10.1016/S0723-0869(04)80009-4 Google Scholar
H. Martini, K. J. Swanepoel, G. Weiss: The geometry of Minkowski spaces - a survey. Part I, Expo. Math. 19 (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6 Google Scholar
J. P. Moreno, R. Schneider: Diametrically complete sets in Minkowski spaces, Isr. J. Math. 191 (2012) 701–720.https://doi.org/10.1007/s11856-012-0003-6 Google Scholar
M. Naszódi, B. Visy: Sets with a unique extension to a set of constant width, in: Discrete Geometry, K. Bezdek (ed.), Marcel Dekker, New York (2003) 373–380.https://doi.org/10.1201/9780203911211.ch27 Google Scholar
R. Payá, Á. Rodríguez-Palacios: Banach spaces which are semi L-summands in their biduals, Math. Ann. 289 (1991) 529–542.https://doi.org/10.1007/BF01446587 Google Scholar
E. S. Polovinkin: Convex bodies of constant width, Dokl. Mat. 70 (2004) 560–562.Google Scholar
M. Spirova: On a theorem of G. D. Chakerian, Contrib. Discrete Math. 5 (2010) 107–118.https://doi.org/10.55016/ojs/cdm.v5i1.62034 Google Scholar
L. Veselý: Convex sets without diametral pairs, Extr. Math. 24 (2009) 271–280.Google Scholar
R. Webster: Convexity, Oxford Science Publications, Oxford Univ. Press, New York (1994).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.

[1] G. Averkov: On cross-section measures in Minkowski spaces, Extr. Math. 18 (2003) 201–208.Google Scholar

[2] M. Baronti, P. L. Papini: Diameters, centers and diametrically maximal sets, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 38 (1995) 11–24.Google Scholar

[3] G. D. Chakerian, H. Groemer: Convex bodies of constant width, in: Convexity and its Applications, P. M. Gruber, J. M Wills (eds.), Birkhäuser, Basel (1983) 49–96.https://doi.org/10.1007/978-3-0348-5858-8_3 Google Scholar

[4] L. Dalla, N. K. Tamvakis: Sets of constant width and diametrically complete sets in normed spaces, Bull. Greek Math. Soc. 26 (1985) 27–39.Google Scholar

[5] C. Franchetti: Relationship between the Jung constant and a certain projection constant in Banach spaces, Ann. Univ. Ferrara, N. Ser., Sez. VII 33 (1977) 39–44.https://doi.org/10.1007/BF02825987 Google Scholar

[6] P. Gritzmannn, V. Klee: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces, Discrete Comput. Geom. 7 (1992) 255–280.https://doi.org/10.1007/BF02187841 Google Scholar

[7] A. S. Granero, J. P. Moreno, R. R. Phelps: Mazur sets in normed spaces, Discrete Comput. Geom. 31 (2004) 411–420.https://doi.org/10.1007/s00454-003-0808-5 Google Scholar

[8] J. P. Moreno, P. L. Papini, R. R. Phelps: Diametrically maximal and constant width sets in Banach spaces, Can. J. Math. 58 (2006) 820–842.https://doi.org/10.4153/CJM-2006-033-1 Google Scholar

[9] J. P. Moreno, P. L. Papini, R. R. Phelps: New families of convex sets related to diametral maximality, J. Convex Analysis 13 (2006) 823–837.https://doi.org/10.68381/jca13061 Google Scholar

[10] H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces - a survey. Part II, Expo. Math. 22 (2004) 93–144.https://doi.org/10.1016/S0723-0869(04)80009-4 Google Scholar

[11] H. Martini, K. J. Swanepoel, G. Weiss: The geometry of Minkowski spaces - a survey. Part I, Expo. Math. 19 (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6 Google Scholar

[12] J. P. Moreno, R. Schneider: Diametrically complete sets in Minkowski spaces, Isr. J. Math. 191 (2012) 701–720.https://doi.org/10.1007/s11856-012-0003-6 Google Scholar

[13] M. Naszódi, B. Visy: Sets with a unique extension to a set of constant width, in: Discrete Geometry, K. Bezdek (ed.), Marcel Dekker, New York (2003) 373–380.https://doi.org/10.1201/9780203911211.ch27 Google Scholar

[14] R. Payá, Á. Rodríguez-Palacios: Banach spaces which are semi L-summands in their biduals, Math. Ann. 289 (1991) 529–542.https://doi.org/10.1007/BF01446587 Google Scholar

[15] E. S. Polovinkin: Convex bodies of constant width, Dokl. Mat. 70 (2004) 560–562.Google Scholar

[16] M. Spirova: On a theorem of G. D. Chakerian, Contrib. Discrete Math. 5 (2010) 107–118.https://doi.org/10.55016/ojs/cdm.v5i1.62034 Google Scholar

[17] L. Veselý: Convex sets without diametral pairs, Extr. Math. 24 (2009) 271–280.Google Scholar

[18] R. Webster: Convexity, Oxford Science Publications, Oxford Univ. Press, New York (1994).Google Scholar