Geodesic
Inscribed and circumscribed polyhedra for a~convex body and continuous functions on a~sphere in Euclidean space
Algebra i analiz, Tome 18 (2006) no. 6, pp. 187-204.

Voir la notice de l'article provenant de la source Math-Net.Ru

Two related problems concerning continuous functions on a sphere $S^{n-1}\subset\mathbb R^n$ are studied, together with the problem of finding a family of polyhedra in $\mathbb R^n$ one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in $\mathbb R^n$. In particular, it is proved that, in every convex body $K\subset\mathbb R^3$, one can inscribe an eight-vertex polyhedron obtained by “equiaugmentation” of a similarity image of any given tetrahedron of class $T$. 
@article{AA_2006_18_6_a2,
     author = {V. V. Makeev},
     title = {Inscribed and circumscribed polyhedra for a~convex body and continuous functions on a~sphere in {Euclidean} space},
     journal = {Algebra i analiz},
     pages = {187--204},
     publisher = {mathdoc},
     volume = {18},
     number = {6},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2006_18_6_a2/}
}
TY  - JOUR
AU  - V. V. Makeev
TI  - Inscribed and circumscribed polyhedra for a~convex body and continuous functions on a~sphere in Euclidean space
JO  - Algebra i analiz
PY  - 2006
SP  - 187
EP  - 204
VL  - 18
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2006_18_6_a2/
LA  - ru
ID  - AA_2006_18_6_a2
ER  - 
%0 Journal Article
%A V. V. Makeev
%T Inscribed and circumscribed polyhedra for a~convex body and continuous functions on a~sphere in Euclidean space
%J Algebra i analiz
%D 2006
%P 187-204
%V 18
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2006_18_6_a2/
%G ru
%F AA_2006_18_6_a2
V. V. Makeev. Inscribed and circumscribed polyhedra for a~convex body and continuous functions on a~sphere in Euclidean space. Algebra i analiz, Tome 18 (2006) no. 6, pp. 187-204. http://geodesic.mathdoc.fr/item/AA_2006_18_6_a2/

[1] Knaster B., “Problem P 4”, Colloq. Math., 1 (1947), 30–31

[2] Floyd E. E., “Real valued mappings of spheres”, Proc. Am. Math. Soc., 6 (1955), 957–959 | DOI | MR | Zbl

[3] Makeev V. V., “O nekotorykh svoistvakh nepreryvnykh otobrazhenii sfer i zadachakh kombinatornoi geometrii”, Geometricheskie voprosy teorii funktsii i mnozhestv, Mezhvuz. mat. sb., Kalinin, 1986, 75–85 | MR | Zbl

[4] Chen W., “Counterexamples to Knaster's conjecture”, Topology, 37 (1998), 401–405 | DOI | MR | Zbl

[5] Kashin B. S., Szarek S. J., “The Knaster problem and the geometry of high-dimensional cubes”, C. R. Acad. Sci. Paris Ser. 1, 336 (2003), 931–936 | DOI | MR | Zbl

[6] Hinrichs A., Richter C., New counterexamples to Knaster's conjecture, Preprint, 2003 | MR

[7] Makeev V. V., “Affinno-vpisannye i affinno-opisannye mnogougolniki i mnogogranniki”, Zap. nauchn. sem. POMI, 231, 1995, 286–298 | MR | Zbl

[8] Makeev V. V., “Trekhmernye mnogogranniki, vpisannye i opisannye vokrug vypuklykh kompaktov”, Algebra i analiz, 12:4 (2000), 1–15 | MR

[9] Makeev V. V., “Trekhmernye mnogogranniki, vpisannye i opisannye vokrug vypuklykh kompaktov, II”, Algebra i analiz, 13:5 (2001), 110–133 | MR | Zbl

[10] Makeev V. V., “O chetyrekhugolnikakh, vpisannykh v zamknutuyu krivuyu”, Matem. zametki, 57:1 (1995), 129–132 | MR | Zbl

[11] Shnirelman L. G., “O nekotorykh geometricheskikh svoistvakh zamknutykh krivykh”, UMN, 1944, no. 10, 34–44

[12] Dyson F. J., “Continuous functions defined on spheres”, Ann. Math., 54 (1951), 534–536 | DOI | MR | Zbl

[13] E. Fadell, S. Husseini, “An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems”, Ergodic Theory Dynamical Systems, 8 (1988), 73–85 | DOI | MR | Zbl

[14] R. T. Z̆ivalevic̆, S. T. Vrec̆ica, “An extension of the ham sandwich theorem”, Bull. London Math. Soc., 22 (1990), 183–186 | DOI | MR

[15] V. Dolnikov, “Transversali semeistv mnozhestv v $\mathbb R^n$ i svyaz mezhdu teoremami Khelli i Borsuka”, Matem. sb., 184:5 (1993), 111–132 | MR

[16] Dolnikov V. L., “Ob odnom obobschenii teoremy o buterbrode”, Matem. zametki, 52:2 (1992), 27–37 | MR

[17] Griffiths H., “The topology of square pegs in round holes”, Proc. London Math. Soc. (3), 62 (1991), 647–672 | DOI | MR | Zbl

[18] Hausel T., Makai E., Szücz A., “Inscribeing cubes and covering by rhombic dodecahedra via equivariant topology”, Mathematica, 47 (2000), 371–397 | MR | Zbl

[19] Kakutani S. A., “A proof, that there exists a circumscribing cube around any bounded closed set in $\mathbb R^3$”, Ann. Math., 43 (1942), 285–303 | MR

[20] Rattray R., “An antipodal point orthogonal point theorem”, Ann. Math., 60 (1954), 502–512 | DOI | MR | Zbl