Pinch's conjecture on $a$-convexity

Abstract

For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\{ 0, 1 \}$ in $\mathbb{R}^{1}$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.