SCHWEITZER 1983 Problem 8

Prove that any identity that holds for every finite $n$ -distributive lattice also holds for the lattice of all convex subsets of the $(n-1)$ -dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $n$ - $\textit{distributive}$ if $$x \wedge (\bigvee_{i=0}^n y_i)=\bigvee_{j=0}^n(x \wedge (\bigvee_{0\leq i \leq n, i \not= j\ }y_i))$$ holds for all elements of the lattice.)

A. Huhn