Computing Min-Convex Hulls in the Affine Building of $$\hbox {SL}_d$$

Abstract

We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of $$\hbox {SL}_d(K)$$ , for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical polytope.