The generalized \(L_p\)-mixed affine surface area
The concept of classical affine surface area was generalized in many ways, in the sense meant here mainly by E. Lutwak (from mixed affine surface area over \(L_p\)-affine surface area to \(L_p\)-mixed affine surface area). Extensions to \(i\)th \(L_p\)-mixed affine surface area followed, where \(i\) is any real number. In the present article, the author studies so-called \((i,j)\)-type \(L_p\)-mixed affine surface area, having Lutwak's notions as subcases. Also, the author lays special emphasize on the case of \((i,-p)\)-type \(L_p\)-mixed affine surface area, establishing the Minkowski inequality and the \(L_p\)-Petty affine projection inequality for this case. Further deep results are obtained, among them also an affirmative answer for the generalized \(L_p\)-Winterniz monotonicity problem.
\(i\)th \(L_p\)-mixed curvature function
\((i,j)\)-type \(L_p\)-mixed affine surface area
scientific article; zbMATH DE number 6505623
scientific article