p-difference: a counterpart of Minkowski difference in the framework of the Lp-Brunn–Minkowski theory

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.