Abstract
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
Highlights
The cube d = [0, 1]d together with the simplex ∆d = conv(0, e1, . . . , ed) and the cross-polytope ♦d = conv(±e1, . . . , ±ed) constitute the Big Three, three infinite families of convex polytopes whose geometric and combinatorial features make them ubiquitous throughout mathematics
The 5-dimensional halfcube was already described by Thomas Gosset [11] in his classification of semi-regular polytopes
Halfcubes appear under the name of demi(hyper)cubes [7] or parity polytopes [26]
Summary
Our name and notation derive from the fact that if S = {k} is a singleton, ∆(d, S) =: ∆(d, k) is the well-known (d, k)-hypersimplex, the convex hull of all vectors v ∈ {0, 1}d with exactly k entries equal to 1 This is a (d − 1)-dimensional polytope for 0 < k < d that makes prominent appearances in combinatorial optimization as well as in algebraic geometry [19]. For appropriate choices of S ⊆ [0, d], we get – the cube d = ∆(d, [0, d]), – the even halfcube Hd = ∆(d, [0, d] ∩ 2Z), – the simplex ∆d = ∆(d, {0, 1}), and – the cross-polytope ∆(d, {1, d − 1}) (up to linear isomorphism). A combinatorial d-cube has the interesting property that all pulling triangulations have the same number of d-dimensional simplices.
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