**4. Conclusions**

Many outstanding works showed that the log-concave function is closely linked to the convex body. This paper presents a new connection between the theory of convex bodies and that of log-concave functions. We study the minimal perimeter of a log-concave function which can be viewed as a functional version of the minimal *L*2 surface area measure of a convex body. A characteristic theorem (Theorem 1) shows that a log-concave function *f* has minimal perimeter if and only if the Borel measure 1*J*(*f*)*μf*(·) is isotropic. The work done in this paper is mainly to propose a special position for log-concave functions and provides a new idea for the study of optimal problems for log-concave functions.

**Author Contributions:** Conceptualization, N.F. and Z.Z.; methodology, N.F. and Z.Z.; formal analysis, N.F. and Z.Z.; investigation, N.F. and Z.Z.; writing—original draft preparation, N.F. and Z.Z.; writing—review and editing, N.F. and Z.Z.; visualization, N.F. and Z.Z.; supervision, N.F. and Z.Z.; project administration, N.F. and Z.Z.; funding acquisition, N.F. and Z.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by China Postdoctoral Science Foundation (No.2019M651001) and the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN201901312).

**Acknowledgments:** The authors would like to thank the reviewers for valuable comments that helped improve the manuscript considerably.

**Conflicts of Interest:** The authors declare no conflict of interest.
