Random triangulations of the d-sphere with minimum volume

We study a higher-dimensional analogue of the Random Travelling Salesman Problem: let the complete d-dimensional simplicial complex Kₙ^d on n vertices be equipped with i. i. d. \ volumes on its facets, uniformly random in 0, 1. What is the minimum volume M₍, ₃ of a sub-complex homeomorphic to the d-dimensional sphere Sᵈ, containing all vertices? We determine the growth rate of M₍, ₂, and prove that it is well-concentrated. For d>2 we prove such results to the extent that current knowledge about the number of triangulations of Sᵈ allows. We remark that this can be thought of as a model of random geometry in the spirit of Angel \& Schramm's UIPT, and provide a generalised framework that interpolates between our model and the uniform random triangulation of Sᵈ.