Convexity has seemingly little to do with fractals, however convex sets with fractal facial structure are not hard to imagine. The convex hull of a spherical Sierpinski triangle shown above is a compact convex set.
Another example is a spherical gasket that comes from the study of limit roots of infinite Coxeter groups. Its facial structure is similar to the compact convex set of symmetric positive semidefinite matrices of dimension 3×3 and trace 1.
One of the most interesting features of the sets obtained as intersections of the semidefinite cone with linear subspaces is that they have large 'gaps' in facial dimensions. In this arxiv preprint we prove that every combination of facial dimensions is possible and talk in generality about convex sets with fractal facial structure.