Geodesic
On Polytopes that are Simple at the Edges
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 36-47.

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

We study some combinatorial properties of polytopes that are simple at the edges. We give an elementary geometric proof of an analog of the hard Lefschetz theorem for the polytopes for which the distance between any two nonsimple vertices is sufficiently large. This implies that the $h$-vector of such polytopes satisfies the relations $h_{[d/2]}\ge h_{[d/2]+1}\ge\cdots\ge h_d=1$, where $d$ is the dimension of the polytope, which proves a special case of Stanley's conjecture. 
@article{FAA_2001_35_3_a3,
     author = {V. A. Timorin},
     title = {On {Polytopes} that are {Simple} at the {Edges}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {36--47},
     publisher = {mathdoc},
     volume = {35},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a3/}
}
TY  - JOUR
AU  - V. A. Timorin
TI  - On Polytopes that are Simple at the Edges
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2001
SP  - 36
EP  - 47
VL  - 35
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a3/
LA  - ru
ID  - FAA_2001_35_3_a3
ER  - 
%0 Journal Article
%A V. A. Timorin
%T On Polytopes that are Simple at the Edges
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2001
%P 36-47
%V 35
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a3/
%G ru
%F FAA_2001_35_3_a3
V. A. Timorin. On Polytopes that are Simple at the Edges. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 36-47. http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a3/

[1] Khovanskii A. G., “Giperploskie secheniya mnogogrannikov, toricheskie mnogoobraziya i diskretnye gruppy v prostranstve Lobachevskogo”, Funkts. analiz i ego pril., 20:1 (1986), 50–61 | MR

[2] Grünbaum B., Convex polytopes, Interscience Publ., London, 1967 | MR | Zbl

[3] Stanley R. P., “The number of faces of a simplicial convex polytope”, Adv. Math., 35 (1980), 236–238 | DOI | MR | Zbl

[4] McMullen P., “On simple polytopes”, Invent. Math., 113 (1993), 419–444 | DOI | MR | Zbl

[5] Timorin V. A., “Analog sootnoshenii Khodzha–Rimana dlya prostykh vypuklykh mnogogrannikov”, UMN, 54:2 (1999), 113–162 | DOI | MR | Zbl

[6] Stanley R. P., “Generalized $H$-vectors, Intersection Cohomology of Toric Varieties, and Related Results”, Adv. Stud. Pure Math., 11 (1987), 187–213 | DOI | MR | Zbl

[7] Khovanskii A. G., “Mnogogranniki Nyutona i toricheskie mnogoobraziya”, Funkts. analiz i ego pril., 11:4 (1977), 56–67 | MR

[8] Danilov V. I., “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2 (1978), 85–134 | MR | Zbl

[9] Fieseler K.-H., “Rational Intersection Cohomology of Projective Toric Varieties”, J. Reine Angew. Math., 413 (1991), 88–98 | MR | Zbl

[10] Bressler P., Lunts V., Intersection cohomology on nonrational polytopes, http://xxx.lanl.gov/abs/math.AG/0002006

[11] Barthel G., Brasselet G.-P., Fieseler K.-H., Kaup L., Combinatorial Intersection Cohomology for Fans, http://xxx.lanl.gov/abs/math.AG/0002181

[12] Pukhlikov A. V., Khovanskii A. G., “Teorema Rimana–Rokha dlya integralov i summ kvazipolinomov po virtualnym mnogogrannikam”, Algebra i analiz, 4:4 (1992), 188–216 | MR | Zbl

[13] Nikulin V. V., “Klassifikatsiya arifmeticheskikh grupp, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR, Ser. Mat., 45 (1981), 113–142 | MR | Zbl