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The aim of this paper is to develop a basic framework of the $L_p$ theory forthe geometry of log-concave functions, which can be viewed as a functional"lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfillthis goal, by combining the $L_p$ Asplund sum of log-concave functions for all$p>1$ and the total mass, we obtain a Pr\'ekopa-Leindler type inequality andpropose a definition for the first variation of the total mass in the $L_p$setting. Based on these, we further establish an $L_p$ Minkowski typeinequality related to the first variation of the total mass and derive avariational formula which motivates the definition of our $L_p$ surface areameasure for log-concave functions. Consequently, the $L_p$ Minkowski problemfor log-concave functions, which aims to characterize the $L_p$ surface areameasure for log-concave functions, is introduced. The existence of solutions tothe $L_p$ Minkowski problem for log-concave functions is obtained for $p>1$under some mild conditions on the pre-given Borel measures.

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem