  In this paper, we introduce and study the locally strongly convex equiaffine
isoparametric hypersurfaces and equiaffine isoparametric functions on the
affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$,
we first introduce the concept of equiaffine parallel hypersurfaces in
$A^{n+1}$, obtaining some fundamental identities with the basic equiaffine
geometric invariants, and then we define the equiaffine isoparametric
hypersurfaces to be ones that are among families of equiaffine parallel
hypersurfaces of constant affine mean curvature in $A^{n+1}$. Finally, we
introduce the concept of equiaffine isoparametric functions on $A^{n+1}$, and
prove that any equiaffine isoparametric hypersurface is exactly a regular level
set of some equiaffine isoparametric function.
