Oriented matroid rigidity of multiplices (Q1580739)

This article concerns a combinatorially-defined class of convex polytopes called ``multiplices'', introduced by the first author in [Mathematika 43, No. 2, 274-285 (1996; Zbl 0874.52007)]. A \(d\)-polytope \(M(d,n+1)\) in \({\mathbb R}^d\) with vertices \(e_0,\ldots,e_n\) is a multiplex if its facets are precisely the convex hulls of \[ e_{i-d+1},\ldots,e_{i-1},e_{i+1},\ldots,e_{i+d-1} \] for \(0\leq i\leq n\). Here \(e_i=e_0\) for \(i<0\) and \(e_i=e_n\) for \(i>n\). Thus the face lattice of a multiplex depends only on \(n\) and \(d\). A simplex is a multiplex with \(n=d\). Just as for simplices, any face or quotient of a multiplex is again a multiplex. The authors show that the full oriented matroid of \(M(d,n+1)\) is uniquely determined by \(n\) and \(d\), provided \(n\leq 2d\). To understand the content of this statement, one may identify the oriented matroid as the set of covectors in \(\{+,-,0\}^{n+1}\) corresponding to partitions of \(e_0,\ldots,e_n\) determined by oriented hyperplanes in \({\mathbb R}^d\). The face lattice is then the subposet of non-negative covectors. That this subposet determines the full oriented matroid is what is meant by ``oriented matroid rigidity.'' There is an inductive construction of multiplices, which is used to produce examples of pairs of multiplices with \(n>2d\) whose underlying (oriented) matroids are distinct, even up to reordering of the vertices.