  An unbiased shift of the two-sided Brownian motion $(B_t \colon t\in{\mathbb
R})$ is a random time $T$ such that $(B_{T+t} \colon t\in{\mathbb R})$ is still
a two-sided Brownian motion. Given a pair $\mu, \nu$ of orthogonal probability
measures, an unbiased shift $T$ solves the embedding problem, if $B_0\sim\mu$
implies $B_{T}\sim\nu$. A solution to this problem was given by Last et al.
(2014), based on earlier work of Bertoin and Le Jan (1992), and Holroyd and
Liggett (2001). In this note we show that this solution minimises ${\mathbb E}
\psi(T)$ over all nonnegative unbiased solutions $T$, simultaneously for all
nonnegative, concave functions $\psi$. Our proof is based on a discrete
concavity inequality that may be of independent interest.
