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.

Differentiability properties of the family of p-parallel bodies

“…In [12] the following counterpart of the p-sum was introduced: for K, E ∈ K n 0 , E ⊆ K, and 1 ≤ p < ∞, the p-difference of K and E is defined as…”

Further inequalities and properties of p-inner parallel bodies

“…As a direct consequence of [10,Proposition 4.2(ii)] one has − + ( + ) ⊆ . Combining these with the definition of -inner parallel body and Minkowski's inequality, we obtain…”

“…Taking into account that lim →− = − (see [10,Proposition 4.3]) and the continuity of the support function with respect to the Hausdorff metric, we have…”

“…Since − + ⊆ (see [10,Proposition 4.2(ii)]), is a tangential body of − + . Moreover, from Lemma 3.4, we get U 0 ( ) = U 0 (…”

“…Let , ∈ K 0 and 1 ≤ < ∞. Then U 0 ( ) ∪ U 0 ( ) ⊆ U 0 ( + ).Let + : R × R → R denote the binary operation which was introduced in[10]: , ) (max{| |, | |} − min{| |, | |} ) sgn 2 : R × R → R is given by sgn = sgn( ) if > 0, sgn( ) if ≤ 0 and | | ≥ | |, sgn( ) if ≤ 0 and | | < | |.Here, sgn denotes the usual sign function. Obviously, this operation satisfies two facts:1.…”