All spaces are assumed to be Tychonoff spaces. Let $X$ be a convex pseudocompact subspace of some locally convex space (LCS).

Question 1. Is it true that the Stone-Cech extension $\beta X$ has the structure of a convex compact set? Is it true that $\beta X$ is homeomorphic to a convex compact subset of some LCS?

The answer to this question is positive if $X=P(Y)$, where $P(Y)$ is the space of probability Radon measures on $X$ in the weak topology [1]. In this case, $Y$ is a pseudocompact space and $\beta P(Y)=P(\beta Y)$. There is a convex compact set $K$ and its dense convex pseudocompact subset $C$ such that $\beta C\neq K$.

Proposition 1. $\beta X$ is path-connected.

This fact is related to the fact that the structure of a convex set with $X$ extends to $\beta X$.

A space $S$ with a (separately) continuous operation $p: [0,1]\times X\times X\to X$ is called a (semi)topological convex set if there is an embedding of $S$ into a linear space (without topology) such that $p(\lambda,x,y)=\lambda x+(1-\lambda)y$.

Theorem 1. If $S$ is a pseudocompact topological convex set, then $\beta S$ is a semitopological convex set.

Clearly, a convex subset of some LCS is a topological convex set. Theorem 1 implies Proposition 1.

Theorem 2. If $S$ is a topological convex set and $S^2$ is pseudocompact, then $\beta S$ is a topological convex set.

Theorem 3. If $S$ is a countable compact semitopological convex set, then $\beta S$ is a semitopological convex set.

The theorems imply that the convex set structure from $S$ extends to $\beta S$.

A (semi)topological convex set $S$ is a universal (semi)topological algebra with continuum operations $p_\lambda: S\times S\to S$, $p_\lambda(x,y)=p(\lambda,x,y)$, where $\lambda\in [0,1]$. The signature of $S$ is continuous, is a segment of $[0,1]$.

The theorems are proved using results on the extension of operations in universal algebras obtained in [2].

[1] Reznichenko, E., “Stone-Cech extensions of probability measure spaces.” arXiv preprint arXiv:2412.11838 (2024).

[2] Reznichenko, E., “Extensions and factorizations of topological and semitopological universal algebras.” Topology and its Applications (2025): 109256.