  The Euclidean first-passage percolation model of Howard and Newman is a
rotationally invariant percolation model built on a Poisson point process. It
is known that the passage time between 0 and $ne_1$ obeys a diffusive upper
bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this
inequality to $Cn/\log n$. The methods follow the strategy used for sublinear
variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a
Bernoulli encoding, but with substantial technical difficulties. To deal with
the different setup of the Euclidean model, we represent the passage time as a
function of Bernoulli sequences and uniform sequences, and develop several
"greedy lattice animal" arguments.
