First-order heat content asymptotics on $\mathsf{R}\mathsf{C}\mathsf{D}(K,N)$ spaces

In this paper, we prove first-order asymptotics on a bounded open set of the heat content when the ambient space is an $\mathsf{R}\mathsf{C}\mathsf{D}(K,N)$ space, under a regularity condition for the boundary that we call measured interior geodesic condition of size $\epsilon$. We carefully study such a condition, relating it to the properties of the disintegration of the signed distance function from $\mathrm{\partial }\mathrm{\Omega }$ studied by Cavalletti and Mondino.