Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1918

Affine Buildings and Tropical Convexity

Michael Joswig·Bernd Sturmfels·Josephine Yu
View original source ↗Read source document ↗Export metadata ↓

Abstract

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.

Explore related subjects

math.MGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Joswig, Bernd Sturmfels, Josephine Yu. 2007-06-13. Affine Buildings and Tropical Convexity. https://arxiv.org/abs/0706.1918

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Divide and Conquer: A Distributed Approach to Five Point Energy Minimization

This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904.

math.MG↗
arXiv2026

Quadri-Figures in Cayley-Klein Planes II: The Miquel-Steiner Theorem

The Miquel-Steiner theorem for a quadrilateral in the euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. The Miquel-Steiner theorem for euclidean planes needs to be slightly modified for Minkowski and galilean planes: Either the circumcircles of the four component triangles touch each other at an isotropic point, or they intersect transversally at an anisotropic point. In elliptic and hyperbolic planes, as well as in dual euclidean and dual Minkowski planes, Miquel-Steiner's theorem does not hold in this form. Instead, a weaker version applies: The circumcircles of the four component triangles of a quadrilateral have a common radical center, which we will also call the Miquel-Steiner point. For specific quadrilaterals (such as cyclic quadrilaterals), the location of the Miquel-Steiner point can be determined more precisely.

math.MG↗
arXiv2026

The Four Color Theorem meets Shapes of Polyhedra

We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $τ$ is parametrized by the set of integer lattice points inside a rational polyhedral convex cone ${\cal C\/}_τ$ of dimension at least 4. There is an integral quadratic form $Q_τ$ on ${\cal C\/}_τ$ whose diagonal part, evaluated on a lattice point, is $3$ times the number of triangles in the corresponding triangulation. We relate this structure to the octahedral stratum of Thurston's moduli space of flat cone structures on the sphere.

math.MG↗