  This article addresses a combinatorial problem with applications to algebraic
geometry. To a convex lattice polytope $P$ and each of its integer dilations
$kP$ one may associate the barycenter of its lattice points. This sequence of
$k$-quantized barycenters converge to the (classical) barycenter of the
polytope considered as a convex body. A basic question arises: is there a
complete asymptotic expansion for this sequence? If so, what are its terms?
This article initiates the study of this question. First, we establish the
existence of such an expansion as well as determine the first two terms.
Second, for Delzant lattice polytopes we use toric algebra to determine all
terms using mixed volumes of virtual rooftop polytopes, or alternatively in
terms of higher Donaldson--Futaki invariants. Third, for reflexive polytopes we
show the quantized barycenters are colinear to first order, and actually
colinear in the case of polygons. The proofs use Ehrhart theory, convexity
arguments, and toric algebra. As applications we derive the complete asymptotic
expansion of the Fujita--Odaka stability thresholds $\delta_k$ on arbitrary
polarizations on (possibly singular) toric varieties. In fact, we show they are
rational functions of $k$ for sufficiently large $k$. This gives the first
general result on Tian's stabilization problem for $\delta_k$-invariants for
(possibly singular) toric Fanos: $\delta_k$ stabilize in $k$ if and only if
they are all equal to $1$, and when smooth if and only if asymptotically Chow
semistable. We also relate the asymptotic expansions to higher
Donaldson--Futaki invariants of test configurations motivated by Ehrhart
theory, and unify in passing previous results of Donaldson, Ono, Futaki, and
Rubinstein--Tian--Zhang.
