  Let $H$ be a Hilbert space and $(\Omega,\mathcal{F},\mu)$ a probability
space. A Hilbert point in $L^p(\Omega; H)$ is a nontrivial function $\varphi$
such that $\|\varphi\|_p \leq \|\varphi+f\|_p$ whenever $\langle f, \varphi
\rangle = 0$. We demonstrate that $\varphi$ is a Hilbert point in $L^p(\Omega;
H)$ for some $p\neq2$ if and only if $\|\varphi(\omega)\|_H$ assumes only the
two values $0$ and $C>0$. We also obtain a geometric description of when a sum
of independent Rademacher variables is a Hilbert point.
