  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 ellipsoid for log-concave function, and establish some
inequalities of the $L_p$ John ellipsoid for log-concave function. Finally, the
analog of Ball's volume ratio inequality for the $L_p$ John ellipsoid of
log-concave function is established.

Functional Analysis;Metric Geometry