17 citations · 73 across the 26 of their papers we have counts for
9 papers · 1 filter
Extension complexity and realization spaces of hypersimplices
Francesco Grande, Arnau Padrol, Raman Sanyal
The (n,k)-hypersimplex is the convex hull of all 0/1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of…
Six topics on inscribable polytopes
Arnau Padrol, Günter M. Ziegler
Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six…
Scribability problems for polytopes
Hao Chen, Arnau Padrol
In this paper we study various scribability problems for polytopes. We begin with the classical -scribability problem proposed by Steiner and generalized by Schulte, which asks…
Universality theorems for inscribed polytopes and Delaunay triangulations
Karim A. Adiprasito, Arnau Padrol, Louis Theran
We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic se…
Polygons as sections of higher-dimensional polytopes
Arnau Padrol, Julian Pfeifle
We show that every heptagon is a section of a -polytope with vertices. This implies that every -gon with can be obtained as a section of a $(2+\lfloor\frac{n}{7…
The universality theorem for neighborly polytopes
Karim A. Adiprasito, Arnau Padrol
In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an even-dimensional neighborly polytope. This in particular…
Neighborly inscribed polytopes and Delaunay triangulations
Bernd Gonska, Arnau Padrol
We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there…
A universality theorem for projectively unique polytopes and a conjecture of Shephard
Karim Alexander Adiprasito, Arnau Padrol
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytop…
Many neighborly polytopes and oriented matroids
Arnau Padrol
In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of ((r+d)^((r/2+d/2)^2))/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the num…