Yes, but similar to the classification of regular solids there are few such polytopes as the dimension gets high enough. There are 5 four dimensional deltatopes and only 3 for each higher dimension (the simplex, the cross-polytope, and the bipyramid over the lower dimensional simplex). This is proved in Sullivan's (unpublished) preprint "Convex Deltatopes in all Dimensions and Polyhedral Soap Films" (available here).
**Abstract** added by J.O'Rourke:
![SullivanAbs][1]
