  We propose the 4D, N=1 supergravitational analogues (avatars) of the 4D, N=1
supersymmetric Born-Infeld action in four dimensions for the first time, by
using superspace. In particular, a new Born-Infeld type generalization of the
Weyl supergravity action is given. A natural new Born-Infeld type
generalization of the Einstein supergravity is found as well. We also brielfy
discuss a construction of the four-dimensional Born-Infeld-Einstein
supergravity from the AdS supergravity in five dimensions, which seems to be
very natural in our approach.
