s-convexity, model sets and their relation

Abstract

The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of parameters s. It is shown that the three Pisot numbers $\frac{1}{2}(1+\sqrt{5})$, $1+\sqrt{2}$, and $2+\sqrt{3}$ related to experimentally observed non-crystallographic symmetries are exceptional with respect to s-convexity.