Arkiv för Matematik

Volume 57 (2019)

Number 2

Pluripotential theory and convex bodies: large deviation principle

Pages: 247 – 283



Turgay Bayraktar (Sabanci University, Istanbul, Turkey)

Thomas Bloom (Department of Mathematics, University of Toronto, Ontario, Canada)

Norman Levenberg (Indiana University, Bloomington, In., U.S.A.)

Chinh H. Lu (Université Paris-Sud, Orsay, France)


We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $(\mathbb{R}^{+})^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P$-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.


convex body, $P$-extremal function, large deviation principle

2010 Mathematics Subject Classification

31C15, 32U15, 32U20

N. Levenberg is supported by Simons Foundation grant No. 354549.

Received 18 August 2018

Received revised 10 February 2019

Accepted 26 February 2019

Published 7 October 2019