Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.