Arnau Padrol

Many Neighborly Polytopes and Oriented Matroids

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of

The universality theorem for neighborly polytopes

{\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}

Enumerating Neighborly Polytopes and Oriented Matroids

((r+d)(r2+d2)2)/(r(r2)2d(d2)2e3r2d2) for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with

Delaunay triangulations with disconnected realization spaces

r+d+1

Polytopes with few vertices and few facets

r+d+1 vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a new simple proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.

Neighborly inscribed polytopes and delaunay triangulations

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of