Geodesic
A Problem of Fejes L. Tóth
Matematičeskie trudy, Tome 5 (2002) no. 1, pp. 102-113 
Yu. G. Nikonorov; N. V. Rasskazova. A Problem of Fejes L. Tóth. Matematičeskie trudy, Tome 5 (2002) no. 1, pp. 102-113. http://geodesic.mathdoc.fr/item/MT_2002_5_1_a6/
@article{MT_2002_5_1_a6,
     author = {Yu. G. Nikonorov and N. V. Rasskazova},
     title = {A~Problem of {Fejes} {L.~T\'oth}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {102--113},
     year = {2002},
     volume = {5},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2002_5_1_a6/}
}
TY  - JOUR
AU  - Yu. G. Nikonorov
AU  - N. V. Rasskazova
TI  - A Problem of Fejes L. Tóth
JO  - Matematičeskie trudy
PY  - 2002
SP  - 102
EP  - 113
VL  - 5
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MT_2002_5_1_a6/
LA  - ru
ID  - MT_2002_5_1_a6
ER  - 
%0 Journal Article
%A Yu. G. Nikonorov
%A N. V. Rasskazova
%T A Problem of Fejes L. Tóth
%J Matematičeskie trudy
%D 2002
%P 102-113
%V 5
%N 1
%U http://geodesic.mathdoc.fr/item/MT_2002_5_1_a6/
%G ru
%F MT_2002_5_1_a6

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

Let $P$ be a convex $n$-gon on the Euclidean plane with edges of lengths $a_1,\dots,a_n$. Denote by $b_i$ the length of the maximal chord of $P$ parallel to $a_i$. For the quantity $\mu(P)=\sum_{i=1}^n{a_i}/{b_i}$, we prove the inequality $3\le\mu(P)\le 4$, which is the Fejes Tóth conjecture. We also give a classification of polygons with $\mu(P)=3$ or $\mu(P)=4$.

[1] Reshetnyak Yu. G., “Odna ekstremalnaya zadacha iz teorii vypuklykh krivykh”, Uspekhi mat. nauk, 8:6 (1953), 125–126 | MR

[2] Khadviger G., Lektsii ob ob'eme, ploschadi poverkhnosti i izoperimetrii, Nauka, M., 1966

[3] Croft N. T., Falconer K. J., Guy R. K., Unsolved Problems in Geometry, Springer-Verlag, Berlin, etc., 1994 | MR | Zbl

[4] Tóth L. Fejes, “Über eine affineinvariante Masszahl bei Eipolyedern”, Studia Sci. Math. Hungar., 5 (1970), 173–180 | MR | Zbl