[math-fun] problem: intersecting annuli

Consider an annulus in Euclidean D-space with inner radius R and outer radius R+W. (If D=2 this means the region inside R+W circle and outside R circle. If D=3 then spheres not circles.) Now consider the set-intersection of D such annuli, respectively centered at the D vertices of a (D-1)-dimensional simplex. What is the AREA (if D=2, or VOLUME if D=3) of this region, as a function of R,W and the simplex? An exact formula should be attainable but complicated when D=2 and D=3. (Trivial answer 2W when D=1.) But I am interested in the LIMIT of large R and small W with simplex fixed, and in this limit there should be a fairly simple asymptotically correct inexact formula. -- Warren D. Smith http://RangeVoting.org