On polytopes simple in edges

TL;DR

This paper explores combinatorial properties of convex polytopes that are simple in edges, proving an analog of the Hard Lefschetz theorem for certain cases and confirming Stanley's conjecture.

Contribution

It introduces a new class of polytopes and proves an analog of the Hard Lefschetz theorem, supporting Stanley's conjecture for these cases.

Findings

Proved an analog of the Hard Lefschetz theorem for specific polytopes

Confirmed Stanley's conjecture for polytopes with well-separated nonsimple vertices

Identified conditions under which combinatorial properties hold for polytopes simple in edges

Abstract

We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's conjecture for such polytopes.