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

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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. 
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     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/}
}
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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/