  Convex algebras, also called (semi)convex sets, are at the heart of modelling
probabilistic systems including probabilistic automata. Abstractly, they are
the Eilenberg-Moore algebras of the finitely supported distribution monad.
Concretely, they have been studied for decades within algebra and convex
geometry.
  In this paper we study the problem of extending a convex algebra by a single
point. Such extensions enable the modelling of termination in probabilistic
systems. We provide a full description of all possible extensions for a
particular class of convex algebras: For a fixed convex subset $D$ of a vector
space satisfying additional technical condition, we consider the algebra of
convex subsets of $D$. This class contains the convex algebras of convex
subsets of distributions, modelling (nondeterministic) probabilistic automata.
We also provide a full description of all possible extensions for the class of
free convex algebras, modelling fully probabilistic systems. Finally, we show
that there is a unique functorial extension, the so-called black-hole
extension.
