The mixed \(L_p\) geominimal surface areas for multiple convex bodies. (English) Zbl 1335.52012
The paper is concerned with some problems in the \(L_p\) Brunn-Minkowski theory of convex bodies. Several mixed \(L_p\) geominimal surface areas are introduced for multiple convex bodies for all \(-n\neq p\in \mathbb R\). Properties like the affine invariance are put forward. Related Alexander-Fenchel-type inequalities, Santaló-type inequalities, affine isoperimetric inequalities, and cyclic inequalities are established. An equivalent formula for the mixed \(p\)-affine surface area is provided. Relations between mixed \(L_p\) geominimal surface areas and the mixed \(p\)-affine surface area are discussed. The \(i\)th mixed \(L_p\) geominimal surface areas and related isoperimetric inequalities are put forward.
Reviewer: Alexey Alimov (Moskva)
MSC:
52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
53A15 | Affine differential geometry |