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The affine L p Pólya–Szegö principle significantly strengthens the classical Pólya–Szegö principle. It is an open problem whether there exists an affine Orlicz Pólya–Szegö principle which includes the affine L p Pólya–Szegö principle as special case. In this paper, an affine Orlicz Pólya–Szegö principle for log-concave functions is established by using functional Steiner symmetrizations. [ABSTRACT FROM AUTHOR]

For sets of finite perimeter, k -codimensional projection bodies are introduced. The corresponding k -codimensional projection inequalities are investigated and their equality cases are characterized. [ABSTRACT FROM AUTHOR]

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz). In order to achieve it, we investigate the graph functions of the given Lipschitz star body K : Along almost all directions u , we can define the graph functions on an open dense subset of the orthogonal projection of K onto u ⊥. [ABSTRACT FROM AUTHOR]