  When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real
dilation parameters $t$, we lose the invariance under integer translations that
exists when $t$ is restricted to be an integer. This paper studies this
phenomenon; in particular, it is shown that, for full-dimensional $P$, not only
there are infinitely many different functions $L_{P + w}(t)$ (for integer $w$),
but that for rational $P$ the collection of these functions identifies $P$
uniquely.
