一句话：刻画凸代数单点扩张，为概率系统终止建模提供完整分类，并给出唯一函子性黑洞扩张。
论文研究凸代数（半凸集）的单点扩张问题。凸代数是有限支撑分布单子的 Eilenberg-Moore 代数，可用于建模概率自动机等概率系统；通过增加一个点，可表示系统终止。作者对一类特定凸代数——向量空间中满足附加技术条件的凸子集 D 的凸子集代数——完整描述了所有可能的扩张；该类包含分布凸子集的凸代数，对应非确定性概率自动机。随后又对自由凸代数（建模完全概率系统）给出所有扩张的完整描述。最后证明存在唯一函子性扩张，即所谓黑洞扩张。
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.