Best constant and value of extremizers for a k-plane transform inequality

TL;DR

This paper determines the best constant and extremizers for the k-plane transform inequality on R^d, resolving a conjecture and extending previous results for specific cases.

Contribution

It explicitly finds extremizers and the optimal constant for the k-plane transform inequality for all k and d, solving a longstanding conjecture.

Findings

Explicit extremizers identified for the inequality.

Exact value of the best constant determined.

Resolved a 1997 conjecture on the k-plane transform.

Abstract

The k-plane transform acting on test functions on R^d satisfies a dilation-invariant L^p to L^q inequality for some exponents p,q. We will explicit some extremizers and the value of the best constant for any value of k and d, solving the limit case of a 1997 conjecture from Baernstein and Loss. This extends their own result for k=2 and Christ's result for k=d-1.