Let
$\tilde\varLambda $
be a polytope in
$\mathbb{R}^d$
,
$\varLambda =\tilde\varLambda \cap\mathbb{Z}^{d}$
be its trace on the group
$\mathbb{Z}^{d}$
and let
$T_{\varLambda }(f)$
be a Toeplitz operator, with a positive symbol
$f\in L^{\infty}(\mathbb{T}^{d})$
, defined on the Hardy space
$H^2(\varLambda )$
. With some additional assumptions on
$f$
, a purely algebraic inverse formula of
$T_{\varLambda }(f)$
is given. The geometric properties of the operators are highlighted when
$\varLambda $
is inflated to get
$\varLambda _\lambda =\lambda \varLambda $
with
$\lambda=2^{m}$
chosen sufficiently large. A localization result for the inverse operator is obtained. When we are sufficiently close to a
$(d-k)$
‐dimensional face of the inflated polytope, the localized inversion formula is reduced to those operators reflecting this proximity. Subsequently an analogue of the strong Szegő limit theorem is stated; a
$d+1$
order asymptotics formula of the trace of the inverse is given that connects each operator of the sum with some geometric measures of our polytope (volume, area,
$\,\ldots$
). Furthermore, the results of Thorsen and Doktorski are retrieved with the help of the Ehrart lattice enumerator formula.
