Geodesic
On construction of bodies of constant width containing a given set
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 215-225
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A survey of the author's results is presented, and the research is continued of the known differential geometry problem on the construction of bodies of constant width containing an arbitrary given bounded set of the same diameter as the width of the bodies. The problem is considered for sets from reflexive Banach spaces in which the unit ball is a generating set. Using earlier results in strongly convex analysis and an explicit formula for constructing one of such bodies of constant width, we establish a criteria for the uniqueness of the complement of an arbitrary set to a body of constant width and propose some algorithms for constructing all bodies of constant width that contain a given set.
Keywords: convex bodies of constant width, strongly convex analysis. 
@article{TIMM_2009_15_4_a17,
     author = {E. S. Polovinkin},
     title = {On construction of bodies of constant width containing a~given set},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {215--225},
     year = {2009},
     volume = {15},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a17/}
}
TY  - JOUR
AU  - E. S. Polovinkin
TI  - On construction of bodies of constant width containing a given set
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2009
SP  - 215
EP  - 225
VL  - 15
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a17/
LA  - ru
ID  - TIMM_2009_15_4_a17
ER  - 
%0 Journal Article
%A E. S. Polovinkin
%T On construction of bodies of constant width containing a given set
%J Trudy Instituta matematiki i mehaniki
%D 2009
%P 215-225
%V 15
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a17/
%G ru
%F TIMM_2009_15_4_a17
E. S. Polovinkin. On construction of bodies of constant width containing a given set. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 215-225. http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a17/

[1] Minkowski H., Geometrie der Zahlen, Teubner, Leipzig–Berlin, 1910, 256 pp. | MR | Zbl

[2] Bonnezen T., Fenkhel V., Teoriya vypuklykh tel, FAZIS, M., 2002, 210 pp.

[3] Dantser L., Gryumbaum B., Kli V., Teorema Khelli i ee primeneniya, Mir, M., 1968, 159 pp. | MR

[4] Chakerian G. D., Groemer H., “Convex bodies of constant width”, Convexity and its Applications, eds. P. M. Gruber, J. M. Wills, Birkhäuser, Basel, 1983, 49–96 | MR

[5] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 470 pp.

[6] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp. | MR | Zbl

[7] Polovinkin E. S., Balashov M. V., Elementy vypuklogo i silno vypuklogo analiza, 2-e izd., FIZMATLIT, M., 2007, 436 pp. | Zbl

[8] Euler L., “De curvis triangularibus”, Acta Acad. Petropol., 2 (1778), 3–30

[9] Minkowski H., “Theorie der konvexen Körper, insbesondere Begündung ihres Oberflächenbegriffs”, Gesammeite Abhandlungen, Vol. 2, Teubner, Leipzig–Berlin, 1911, 131–229

[10] Minkowski H., “O telakh postoyannoi shiriny”, Mat. sb., 25:3 (1905), 505–508 ; Gesammeite Abhandlungen, Vol. 2, Teubner, Leipzig–Berlin, 1911, 277–279 | Zbl

[11] Blaschke W., Hessenberg G., “Lehsätze über konvexe Körper”, Jber. Deutsch. Math. Vereinig, 26 (1917), 215–220 | Zbl

[12] Lebesgue H., “Sur quelques questions de minimum, relatives aux courbes orbiformes, et sur leurs rapports avec le calcul des variations”, J. Math. Pures Appl., 4:8 (1921), 67–96 | Zbl

[13] Jessen B., “Über konvexe Punktmengen konstanter Breite”, Math. Z., 29 (1928), 378–380 | Zbl

[14] Eggleston H. G., “Sets of constant width in finite dimentional Banach spaces”, Israel J. Math., 3 (1965), 163–172 | DOI | MR | Zbl

[15] Karasev R. N., “O kharakterizatsii porozhdayuschikh mnozhestv”, Modelirovanie i analiz informatsionnykh sistem (Yaroslavl), 8:2 (2001), 3–9

[16] Polovinkin E. S., “O silno vypuklykh mnozhestvakh i silno vypuklykh funktsiyakh”, Itogi nauki i tekhniki. Sovremennaya matematika i ee prilozheniya. Temat. obz., 61, Izd-vo VINITI, M., 1999, 66–138 | MR | Zbl

[17] Polovinkin E. S., “O telakh postoyannoi shiriny”, Dokl. RAN, 397:3 (2004), 313–315 | MR

[18] Polovinkin E. S., Sidenko S. V., “Dopolnenie mnozhestv do tel postoyannoi shiriny”, Fiz.-mat. nauki, Uchen. zapiski Kazanskogo gos. un-ta, 148, no. 2, 2006, 132–143 | Zbl

[19] Polovinkin E. S., Sidenko S. V., “Dopolnenie tetraedra do tela postoyannoi shiriny”, Nekotorye problemy fundamentalnoi i prikladnoi matematiki i ikh prilozheniya v zadachakh fiziki, mezhduved. sb. st., MFTI, M., 2005, 184–198