P-difference: a counterpart of Minkowski difference in the framework of the $$L_p$$ L p -Brunn–Minkowski theory

Abstract

As a substraction counterpart of the well-known p-sum of convex bodies, we introduce the notion of p-difference. We prove several properties of the p-difference, introducing also the notion of p-(inner) parallel bodies. We prove an analog of the concavity of the family of classical parallel bodies for the p-parallel ones, as well as the continuity of this new family, in its definition parameter. Further results on inner parallel bodies are extended to p-inner ones; for instance, we show that tangential bodies are characterized as the only convex bodies such that their p-inner parallel bodies are homothetic copies of them.