The endomorphisms of the lattice of closed convex cones (Q945967)

Let \({\mathcal C}^d\) denote the set of (closed convex) cones in \({\mathbb R}^d\). For \(A, B \in {\mathcal C}^d\), we define \(A \vee B\) to be the closure of the convex hull of \(A \cup B\). Thus we can view \({\mathcal C}^d\) as a lattice with the operations \(\cap\) and \(\vee\). The main theorem of the paper determines all endomorphisms of this lattice. More precisely, for \(d \geq 3\), if \(\phi: {\mathcal C}^d \to {\mathcal C}^d\) satisfies \[ \phi(A \cap B) = \phi(A) \cap \phi(B) \qquad \text{ and } \qquad \phi(A \vee B) = \phi(A) \vee \phi(B) \] for all \(A, B \in {\mathcal C}^d\), then either \(\phi\) is constant or there exists a linear transformation \(g \in GL(d)\) such that \(\phi(C) = gC\) for all \(C \in {\mathcal C}^d\).