The Lp analogues of the Petty projection inequality and the Busemann-Petty centroid inequality are established.An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.This article deals with affine isoperimetric inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke to Dupin (see e.g., the books of Schneider [32] and Leichtweiß [17] for references). If K is an origin-symmetric convex body in Euclidean nspace, R n , then the centroid body of K is the body whose boundary consists of the locus of the centroids of the halves of K formed when K is cut by codimension 1 subspaces. Blaschke (see Schneider [32] for references) conjectured that the ratio of the volume of a body to that of its centroid body attains its maximum precisely for ellipsoids. This conjecture was proven by Petty [27] who also extended the definition of centroid bodies and gave centroid bodies their name. When written as an inequality, Blaschke's conjecture is known as the Busemann-Petty centroid inequality. Busemann's name is attached to the inequality because Petty showed that Busemann's random simplex inequality ([5]) could be reinterpreted as what would become known as the Busemann-Petty centroid inequality. In recent times, centroid bodies (and their

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Both ionic liquids and water are typical green solvents. In this work, the phase behavior of the ternary system consisting of ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate (bmimPF6), TX-100, and water was determined at 25.0 degrees C. The water-in-bmimPF6, bicontinuous, and bmimPF6-in-water microregions of the microemulsions were identified by cyclic voltammetry method using potassium ferrocyanide K4Fe(CN)6 as the electroactive probe. Dynamic light scattering (DLS) and the UV-vis method were used to characterize the microemulsions. It was demonstrated that the hydrodynamic diameter (Dh) of the bmimPF6-in-water microemulsions is nearly independent of the water content but increases with increasing bmimPF6 content due to the swelling of the micelles by the ionic liquid. The UV-vis further confirmed the existence of water domains in the water-in-bmimPF6 microemulsions, and the salt potassium ferricyanide K3Fe(CN)6 could be dissolved in the water domains.

Abstract. In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

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We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to n independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also clarify a small gap in the known argument classifying the equality case of the log-Brunn-Minkowski conjecture for unconditional convex bodies. *

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