532 ON TRIANGULATIONS OF THE 3-BALL AND THE SOLID TORUS G. Bohus, W. Jockusch, C. Lee, N. Prabhu, November 1990 We show that neither the 3-ball nor the solid torus admits a triangulation in which (i) every vertex is on the boundary, and (ii) every tetrahedron has exactly one triangle on the boundary. (Such triangulations are relevant to an unresolved conjecture of Perles.) Our result settles a question posed at the DIMACS Workshop on Polytopes and Convex Sets.