The diameter of type D associahedra and the non-leaving-face property

Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in connection to finite type cluster algebras. Following recent work of L. Pournin in types $A$ and $B$, this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the $n$-dimensional associahedron of type $D$ is precisely $2n-2$ for all $n$ greater than $1$. Furthermore, we show that all type $BCD$ associahedra have the non-leaving-face property, that is, any geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. This property was already proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type $A$. In contrast, we present relevant examples related to the associahedron that do not always satisfy this property.Comment: 18 pages, 14 figures. Version 3: improved presentation, simplification of Section 4.1. Final versio

Computational Aspects of the Hausdorff Distance in Unbounded Dimension

We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed to be homothetically transformed in order to minimize its Hausdorff distance to another one. For this problem, we characterize optimal solutions, deduce a Helly-type theorem and give polynomial time (approximation) algorithms for polytopes

Cluster algebras of type D: pseudotriangulations approach

We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as well as for quivers and their mutations. We also present a new combinatorial interpretation of cluster variables in terms of perfect matchings of a graph after deleting two of its vertices. This interpretation differs from known interpretations in the literature. Its main feature, in contrast with other interpretations, is that for a fixed initial cluster seed, one or two graphs serve for the computation of all cluster variables. Finally, we discuss applications of our model to polytopal realizations of type $D$ associahedra and connections to subword complexes and $c$-cluster complexes.Comment: 21 pages, 21 figure

Geometric Permutations of Non-Overlapping Unit Balls Revisited

Given four congruent balls $A, B, C, D$ in $R^{d}$ that have disjoint interior and admit a line that intersects them in the order $ABCD$, we show that the distance between the centers of consecutive balls is smaller than the distance between the centers of $A$ and $D$. This allows us to give a new short proof that $n$ interior-disjoint congruent balls admit at most three geometric permutations, two if $n\ge 7$. We also make a conjecture that would imply that $n\geq 4$ such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example of a highly degenerate nature