Optimal $L^2$-extensions on tube domains and a simple proof of Pr\'ekopa's theorem

TL;DR

This paper establishes the optimal $L^2$-extension theorem on tube domains and uses it to provide a straightforward proof of Prékopa's theorem, connecting complex analysis with convexity properties.

Contribution

It introduces an optimal $L^2$-extension result on tube domains and offers a simplified proof of Prékopa's theorem, enhancing understanding of complex-analytic and convexity relations.

Findings

Proved the optimal $L^2$-extension theorem on tube domains.

Provided a simple proof of Prékopa's theorem using the extension theorem.

Strengthened the link between complex analysis and convexity principles.

Abstract

We prove the optimal $L^2$-extension theorem of Ohsawa-Takegoshi type on a tube domain. As an application, we give a simple proof of Pr\'ekopa's theorem.