Geodesic
Diameters in Typical Convex Bodies
Canadian journal of mathematics, Tome 42 (1990) no. 1, pp. 50-61

Voir la notice de l'article provenant de la source Cambridge University Press

The most usual diameters in the world are those of a sphere and they all contain its centre. More generally, a chord of a convex body in R d is called a diameter if there are two parallel supporting hyperplanes at the two endpoints of the chord.It is easily seen that there are points on at least two diameters. From a result of Kosiński [6] proved in a more general setting it follows that every convex body has a point lying on at least three diameters. Does a typical convex body behave like a sphere and contain a point on infinitely or even uncountably many diameters?But what is a typical convex body? The space K of all convex bodies (d-dimensional compact convex sets) in R d , equipped with the Hausdorff metric, is a Baire space. 
Bárány, Imre; Zamfirescu, Tudor. Diameters in Typical Convex Bodies. Canadian journal of mathematics, Tome 42 (1990) no. 1, pp. 50-61. doi: 10.4153/CJM-1990-003-8
@article{10_4153_CJM_1990_003_8,
     author = {B\'ar\'any, Imre and Zamfirescu, Tudor},
     title = {Diameters in {Typical} {Convex} {Bodies}},
     journal = {Canadian journal of mathematics},
     pages = {50--61},
     year = {1990},
     volume = {42},
     number = {1},
     doi = {10.4153/CJM-1990-003-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1990-003-8/}
}
TY  - JOUR
AU  - Bárány, Imre
AU  - Zamfirescu, Tudor
TI  - Diameters in Typical Convex Bodies
JO  - Canadian journal of mathematics
PY  - 1990
SP  - 50
EP  - 61
VL  - 42
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1990-003-8/
DO  - 10.4153/CJM-1990-003-8
ID  - 10_4153_CJM_1990_003_8
ER  - 
%0 Journal Article
%A Bárány, Imre
%A Zamfirescu, Tudor
%T Diameters in Typical Convex Bodies
%J Canadian journal of mathematics
%D 1990
%P 50-61
%V 42
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1990-003-8/
%R 10.4153/CJM-1990-003-8
%F 10_4153_CJM_1990_003_8

[1] 1. Besicovitch, A.S. and Zamfirescu, T., On pencils of diameters in convex bodies, Rev. Roum. Math. Pures Appl. 11 (1966), 637–639. Google Scholar

[2] 2. Hammer, P.C., Problem 14 in colloquium on Convexity, Copenhagen (1965). Google Scholar

[3] 3. Hammer, P.C. and Sobczyk, A., Planar line families II, Proc. Amer. Math. Soc. 4 (1953), 341–349. Google Scholar

[4] 4. Heil, E., Concurrent normals and critical points under weak smoothness assumptions, Ann. New York Acad. Sci. 440 (1985), 170–178. Google Scholar

[5] 5. Klee, V., Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51–63. Google Scholar

[6] 6. Kosiński, A., On a problem of Steinhaus, Fund. Math. 46 (1958), 47–59. Google Scholar

[7] 7. Zamfirescu, T., Most convex mirrors are magic, Topology 21 (1982), 65–69. Google Scholar

[8] 8. Zamfirescu, T., Points on infinitely many normals to convex surfaces, J. Reine Angew. Math. 350 (1984), 183–187. Google Scholar

[9] 9. Zamfirescu, T., Intersecting diameters in convex bodies, Ann. Discrete Math. 20 (1984), 311–316. Google Scholar

[10] 10. Zamfirescu, T., Using Baire categories in geometry, Rend. Sem. Math. Univ. Politecn. Torino 43 (1985), 67–88. Google Scholar

Cité par Sources :