The aim of this paper is to develop the $L_p$ John ellipsoid for the geometry of log-concave functions. Using the results of the $L_p$ Minkowski theory for log-concave function established in cite{fan-xin-ye-geo2020}, we characterize the $L_p$ John e
llipsoid for log-concave function, and establish some inequalities of the $L_p$ John ellipsoid for log-concave function. Finally, the analog of Balls volume ratio inequality for the $L_p$ John ellipsoid of log-concave function is established.
In this paper, the functional Quermassintegrals of log-concave functions in $mathbb R^n$ are discussed, we obtain the integral expression of the $i$-th functional mixed Quermassintegrals, which are similar to the integral expression of the $i$-th Quermassintegrals of convex bodies.
