A proof of an extension of the icosahedral conjecture of Steiner for generalized deltahedra

In this note we introduce a new family of convex polyhedra which we call the family of generalized deltahedra or in short, the family of g-deltahedra. Here a g-deltahedron is a convex polyhedron in Euclidean 3-space whose each face is an edge-to-edge union of some triangles each being congruent to a given regular triangle. Steiner’s famous icosahedral conjecture (1841) says that among all convex polyhedra isomorphic to an icosahedron (that is having the same face structure as an icosahedron) the regular icosahedron has the smallest isoperimetric quotient. In this paper we prove that the regular icosahedron has the smallest isoperimetric quotient among all g-deltahedra.