|
Some extremal problems for vector bundles
Author(s):
V.
V.
Makeev
Translated by:
B. M. Bekker
Original publication:
Algebra i Analiz,
tom 19
(2007),
nomer 2.
Journal:
St. Petersburg Math. J.
19
(2008),
261-277.
MSC (2000):
Primary 52A99, 51M20
Posted:
February 7, 2008
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Analogs of well-known problems and theorems on convex bodies are considered for the case where the convex bodies are replaced by continuous fields of convex bodies in vector bundles.
References:
-
- 1.
- V. V. Makeev, On some combinatorial geometry problems for vector bundles, Algebra i Analiz 14 (2002), no. 6, 169-191; English transl., St. Petersburg Math. J. 14 (2003), no. 6, 1017-1032. MR 1965917 (2005b:52009)
- 2.
- B. Grünbaum, Measures of symmetry for convex sets, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, RI, 1963, pp. 233-270; Borsuk's problem and related questions, ibid. pp. 271-284. MR 0156259 (27:6187); MR 0154183 (27:5134)
- 3.
- V. V. Makeev, Planar sections of convex bodies and universal fibrations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), 219-233; English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 249-256. MR 1879268 (2002k:52004)
- 4.
- I. Fáry, Sur la densité des réseaux de domaines convexes, Bull. Soc. Math. France 78 (1950), 152-161. MR 0039288 (12:526a)
- 5.
- L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Grundlehren Math. Wiss. in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 65, Springer-Verlag, Berlin, 1953. MR 0057566 (15:248b)
- 6.
- A. Besicovitch, Measure of asymmetry of convex curves, J. London Math. Soc. 23 (1948), 237-240. MR 0027543 (10:320r)
- 7.
- L. G. Shnirel'man, On certain geometrical properties of closed curves, Uspekhi Mat. Nauk Vyp. 10 (1944), 34-44. (Russian) MR 0012531 (7:35c)
- 8.
- V. V. Makeev, The Knaster problem and almost spherical sections, Mat. Sb. 180 (1989), no. 3, 424-431; English transl., Math. USSR-Sb. 66 (1990), no. 2, 431-438. MR 0993234 (90d:55005)
- 9.
- -, Approximation of two-dimensional sections of convex bodies by disks and ellipses, Algebra i Analiz 16 (2004), no. 6, 162-171; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1043-1049. MR 2117452 (2005i:52008)
- 10.
- -, Affine-inscribed and affine-circumscribed polygons and polytopes, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 231 (1995), 286-298; English transl., J. Math. Sci. (N.Y.) 91 (1998), no. 6, 3518-3525. MR 1434300 (98b:52004)
- 11.
- A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Acad. Press, Jerusalem, 1961, pp. 123-160. MR 0139079 (25:2518)
- 12.
- V. V. Makeev, On geometric properties of three-dimensional convex bodies, Algebra i Analiz 14 (2002), no. 5, 96-109; English transl., St. Petersburg Math. J. 14 (2003), no. 5, 781-790. MR 1970335 (2004c:52001)
- 13.
- -, On approximation of a three-dimensional convex body by cylinders, Algebra i Analiz 17 (2005), no. 2, 133-144; English transl., St. Petersburg Math. J. 17 (2006), no. 2, 315-323. MR 2159587 (2006e:52007)
- 14.
- R. Buck and E. Buck, Equipartition of convex sets, Math. Mag. 22 (1949), 195-198. MR 0029521 (10:621c)
- 15.
- V. V. Makeev, Some properties of continuous mappings of spheres and problems in combinatorial geometry, Geometric Questions in the Theory of Functions and Sets, Kalinin. Gos. Univ., Kalinin, 1986, pp. 75-85. (Russian) MR 1027885
- 16.
- Yu. G. Reshetnyak, An extremal problem from the theory of convex curves, Uspekhi Mat. Nauk (N.S.) 8 (1953), no. 6, 125-126. (Russian) MR 0061400 (15:819d)
- 17.
- D. Laugwitz, Konvexe Mittelpunktsbereiche und normierte Räume, Math. Z. 61 (1954), 235-244. MR 0066667 (16:613d)
- 18.
- V. V. Makeev, On some geometric properties of convex bodies. II, Algebra i Analiz 15 (2003), no. 6, 74-85; English transl., St. Petersburg Math. J. 15 (2004), no. 6, 867-874. MR 2044632 (2004k:52021)
- 19.
- F. J. Dyson, Continuous functions defined on spheres, Ann. of Math. (2) 54 (1951), 534-536. MR 0044620 (13:450f)
- 20.
- J. Pal, Über ein elementares Variationsproblem, Danske Vid. Selsk Mat.-Fys. Medd. 3 (1920), no. 2, 35 p.
- 21.
- V. V. Makeev, Universal coverings. I, Ukrain. Geom. Sb. No. 24 (1981), 70-79. (Russian) MR 0629813 (83e:52009)
- 22.
- D. Gale, On inscribing
-dimensional sets in a regular -simplex, Proc. Amer. Math. Soc. 4 (1953), 222-225. MR 0053534 (14:787b) - 23.
- V. V. Makeev, The asphericity of the shadows of a convex body, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 329 (2005), 67-78; English transl., J. Math. Sci. (N. Y.) 140 (2007), no. 4, 535-541. MR 2215332 (2007a:52004)
- 24.
- E. Fadell and S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems 8 (1988), Charles Conley Memorial Issue, 73-85. MR 0967630 (89k:55002)
- 25.
- R. T. Zivaljević and S. T. Vrećica, An extension of the ham sandwich theorem, Bull. London Math. Soc. 22 (1990), 183-186. MR 1045292 (91j:52006)
- 26.
- V. L. Dol'nikov, Transversals of families of sets in
and a relationship between Helly and Borsuk theorems, Mat. Sb. 184 (1993), no. 5, 111-132; English transl., Russian Acad. Sci. Sb. Math. 79 (1994), no. 1, 93-107. MR 1239754 (94g:52005) - 27.
- F. Cohen and E. L. Lusk, Configuration-like spaces and the Borsuk-Ulam theorem, Proc. Amer. Math. Soc. 56 (1976), 313-317. MR 0425949 (54:13899)
- 28.
- A. I. Shcherba, On estimate of the perimeter of a norming figure in the Minkowski plane, 5 Internat. Conf. on Geometry and Topology in Memory A. V. Pogorelov (Cherkassy, 2003): Thesis. (Russian)
- 29.
- -, On estimate of the perimeter of a unit disk in the Minkowski plane, Trudy Rubtsov. Industr. Inst. Vyp. 12 (2003), 96-107. (Russian)
- 30.
- V. V. Makeev, An upper bound for the perimeter of a nonsymmetric unit disk in the Minkowski plane, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 299 (2003), 262-266; English transl., J. Math. Sci. (N.Y.) 131 (2005), no. 1, 5406-5408. MR 2038827 (2005a:52007)
Similar Articles:
Retrieve articles in St. Petersburg Mathematical Journal
with MSC
(2000):
52A99, 51M20
Retrieve articles in all Journals with MSC
(2000):
52A99, 51M20
Additional Information:
V.
V.
Makeev
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email:
mvv57@inbox.ru
DOI:
10.1090/S1061-0022-08-00998-9
PII:
S 1061-0022(08)00998-9
Keywords:
Field of convex bodies,
mass distribution,
normed plane
Received by editor(s):
14/FEB/2005
Posted:
February 7, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
|