On Some Concepts Related to Star-Shaped Sets

The convenient theoretical properties of the support function and the Minkowski addition-based arithmetic have been shown to be useful when dealing with compact and convex sets on Rp. However, both concepts present several drawbacks in certain contexts. The use of the radial function instead of the support function is suggested as an alternative to characterize a wider class of sets—the so-called star-shaped sets—which contains the class of compact and convex sets as a particular case. The concept of random star-shaped set is considered, and some statistics for this kind of variable are shown. Finally, some measures for comparing star-shaped sets are introduced.