Convex deltahedra in higher dimensions

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 This is proved in Sullivan's (unpublished) preprint "Convex Deltatopes in all Dimensions and Polyhedral Soap Films" (available here).

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Abstract added by J.O'Rourke: