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In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar o-symmetric convex bodies. In this paper, we give a positive answer to Dar's conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality.For planar o-symmetric convex bodies, the log-Brunn-Minkowski inequality was established by Böröczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn-Minkowski inequality, for planar o-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar's conjecture, especially, the definition of "dilation position", inspires us to obtain a general version of the log-Brunn-Minkowski inequality. As expected, this inequality implies the classical Brunn-Minkowski inequality for all planar convex bodies.

Abstract. Using shadow systems, we provide a new proof of the Orlicz Busemann-Petty centroid inequality, which was first obtained by Lutwak, Yang and Zhang.

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.

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Hölder's inequality states that x p y q − −x, y ≥ 0 for any (x, y) ∈ L p (Ω) × L q (Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y| q−2 : x p y q − −x, y ≥ (A similar result holds for complex valued functions with Re(x, y) substituting for x, y. We obtain these inequalities from some stronger (though slightly more involved) ones.