Let α1, α2, . . . , αn, β1, β2, . . . , βn be strictly positive reals with α1 +α2 + · · ·+αn = β1 + β2 + · · ·+ βn = s. In this paper, the inequality |||Aα1Bβ1Aα2 · · ·AαnBβn ||| ≤ |||AB|||s when A and B are positive-definite matrices is studied. Related questions are also studied.

The norm defined by Busemann’s inequality establishes a class of star body intersection body. This class of star body plays a key role in the solution of Busemann-Petty problem. In 2003, Giannapoulos [1] defined a norm for a new class of half-section. Based on this norm, we give a geometric generalization of Busemann-Petty problem, and get its answer as a result.

Often, when several norms are present and may be in conflict, individuals will display a self-serving bias, privileging the norm that best serves their interests. Xiao and Bicchieri (J Econ Psychol 31(3):456–470, 2010) tested the effects of inequality on reciprocating behavior in trust games and showed that—when inequality increases—reciprocity loses its appeal. They hypothesized that self-serv...

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function f ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on...