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 Kₙᵈ on n vertices be equipped with i.i.d.\ volumes on its facets, uniformly random in $[0,1]$. What is the minimum volume Mn,d of a sub-complex homeomorphic to the d-dimensional sphere Sᵈ, containing all vertices? We determine the growth rate of Mn,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.