@article{19538b9d677944dbaad4f1b561c76fb0,

title = "The equality cases of the Ehrhard–Borell inequality",

abstract = "The Ehrhard–Borell inequality is a far-reaching refinement of the classical Brunn–Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn–Minkowski theory, the equality cases in this inequality are far from evident from the known proofs. The equality cases are settled systematically in this paper. An essential ingredient of the proofs are the geometric and probabilistic properties of certain degenerate parabolic equations. The method developed here serves as a model for the investigation of equality cases in a broader class of geometric inequalities that are obtained by means of a maximum principle.",

keywords = "Degenerate parabolic equations, Ehrhard–Borell inequality, Equality cases, Gaussian Brunn–Minkowski inequalities, Gaussian measures",

author = "Yair Shenfeld and {van Handel}, Ramon",

note = "Funding Information: This work was supported in part by NSF grant CAREER-DMS-1148711 and by the ARO through PECASE award W911NF-14-1-0094 . The paper was completed while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140 . The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis and Applications is gratefully acknowledged. Finally, we thank a referee for helpful comments that improved the presentation of this paper. Funding Information: This work was supported in part by NSF grant CAREER-DMS-1148711 and by the ARO through PECASE award W911NF-14-1-0094. The paper was completed while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis and Applications is gratefully acknowledged. Finally, we thank a referee for helpful comments that improved the presentation of this paper. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = jun,

day = "20",

doi = "10.1016/j.aim.2018.04.013",

language = "English (US)",

volume = "331",

pages = "339--386",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

}