Stability for a strengthened Borell–Brascamp–Lieb inequality

ABSTRACT

We study stability issues for some integral inequalities, in particular for the extremal case of the so-called Borell–Brascamp–Lieb inequalities. In this case, when near equality is realized in dimension one, we prove that the involved functions must be $L^1$-close to be quasiconcave. Further results about equality conditions in some other cases are provided.

KEYWORDS:

• Geometric-analytic inequalities
• stability
• Borell–Brascamp–Lieb inequalities
• Brunn–Minkowski inequality

AMS Subject Classifications:

• 26D15
• 52A40

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors have been partially supported by INdAM in the framework of a GNAMPA project, by Università di Firenze with a ‘Progetto Strategico di Ateneo 2015’ and by MIUR in the framework of a PRIN 2013 project.