On Burkholder’s biconvex-function characterization of Hilbert spaces

Abstract:

Suppose that ${\mathbf {X}}$ is a real or complex Banach space with norm $| \cdot |$. Then ${\mathbf {X}}$ is a Hilbert space if and only if \[ E|x + Y| \geqslant 1\] for all $x \in {\mathbf {X}}$ and all ${\mathbf {X}}$-valued Bochner integrable functions $Y$ on the Lebesgue unit interval satisfying $EY = 0$ and $|Y| \geqslant 1$ a.e. This leads to a simple proof of the biconvex-function characterization due to Burkholder.