Convexity of Lp-intersection bodies

Intersection Bodies of Polytopes: Translations and Convexity

“…Furthermore, an intersection body can be both convex and non-convex. Convexity is certified Busemann's theorem [Bus49], which states that IK is convex if K is a convex body centered at the origin (i.e., K is centrally symmetric, where the center of symmetry is the origin), and this statement has been generalized to L p -intersection bodies [Ber09]. On the other hand, given a convex body K ⊆ R d , there always exists some t ∈ R d such that I(K + t) is not convex [Gar06,Thm.…”

The Dual Orlicz–Aleksandrov–Fenchel Inequality

“…Let S n denote the set of star bodies about the origin in R n . The radial addition and volume are the core and essence of the classical dual Brunn-Minkowski theory and played an important role in the theory (see, e.g., [20,[37][38][39][40][41][42] for recent important contributions). Lutwak [43] introduced the concept of dual mixed volumes that laid the foundation of the dual Brunn-Minkowski theory.…”

“…The operation of the L p -harmonic radial addition and L p -dual Minkowski, Brunn-Minkwski inequalities are the basic concept and inequalities in the L p -dual Brunn-Minkowski theory. The latest information and important results of this theory can be referred to [32,37,39,40,[47][48][49][50][51] and the references therein. For a systematic investigation on the concepts of the addition for convex body and star body, we refer the reader to [26,48,50].…”