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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.

The symmetry of convex bodies of constant width is discussed in this paper. We proved that for any convex body K ⊂ R n of constant width, 1 ≤ as, where as ∞ (·) denotes the Minkowski measure of asymmetry for convex bodies. Moreover, the equality holds on the left-hand side precisely iff K is an Euclidean ball and the upper bounds are attainable, in particular, if n = 3, the equality holds on the right-hand side if K is a Meissner body.

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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.