The equality cases of the Ehrhard–Borell inequality

Yair Shenfeld, Ramon van Handel

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)339-386
Number of pages48
JournalAdvances in Mathematics
StatePublished - Jun 20 2018

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Degenerate parabolic equations
  • Ehrhard–Borell inequality
  • Equality cases
  • Gaussian Brunn–Minkowski inequalities
  • Gaussian measures


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