Improved $L_p-$mixed volume inequality for convex bodies

A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csiszár-Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the $L_p-$Brunn-Minkowski inequality is also proved as a corollary.

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