  The pentagram map on polygons in the projective plane was introduced by R.
Schwartz in 1992 and is by now one of the most popular and classical discrete
integrable systems. In the present paper we introduce and prove integrability
of long-diagonal pentagram maps on polygons in $\mathbb{R}\mathrm{P}^d$,
encompassing all known integrable cases. We also establish an equivalence of
long-diagonal and bi-diagonal maps and present a simple self-contained
construction of the Lax form for both. Finally, we prove the continuous limit
of all these maps is equivalent to the $ (2,d+1)$-KdV equation, generalizing
the Boussinesq equation for $d=2$.
