Abstract. In 1996, E. Lutwak extended the important concept of geominimal surface area to L p version, which serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative differential geometry, and Minkowskian geometry. In this paper, by using the concept of Orlicz mixed volume, we extend geominimal surface area to the Orlicz version and give some properties and an isoperimetric inequalities for the Orlicz geominimal surface areas.Mathematics subject classification (2010): 52A39, 52A40.

In this article, a classification of continuous, linearly intertwining, symmetric L p L_p -Blaschke ( p > 1 p>1 ) valuations is established as an extension of Haberl’s work on Blaschke valuations. More precisely, we show that for dimensions n ≥ 3 n \geq 3 , the only continuous, linearly intertwining, normalized symmetric L p L_p -Blaschke valuation is the normalized L p L_p -curvature image operator, while for dimension n = 2 n = 2 , a rotated normalized L p L_p -curvature image operator is the only additional one. One of the advantages of our approach is that we deal with normalized symmetric L p L_p -Blaschke valuations, which makes it possible to handle the case p = n p=n . The cases where p ≠ n p \not =n are also discussed by studying the relations between symmetric L p L_p -Blaschke valuations and normalized ones.

In this paper, the authors define a harmonic Orlicz combination and a dual Orlicz mixed volume of star bodies, and then establish the dual Orlicz-Minkowski mixedvolume inequality and the dual Orlicz-Brunn-Minkowksi inequality.

In this paper, some dual Brunn-Minkowski inequalities are established for intersection bodies for the harmonic Blaschke additions and p-radial additions.

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