The "shrine problem" is, given a family S of shapes (let's say certain subsets of R^n) for which it makes sense to talk about a continuous curve [0,1] -> F, find a selection function P: S -> R^n such that for all s in S, P(s) is a member of s. P must also satisfy two conditions: 1) Given a continuous curve of shapes s:[0,1] -> S, the map [0,1] -> R^n given by t |-> P(s_t) is continuous, and 2) If some isometry I: R^n -> R^n carries one shape s_1 of S onto another one s_2, then P(s_2) = I(P(s_1)). --------------------------------------------------------------------------------------------------- Case in point: a) All embedded closed arcs in the plane. If we cared only about rectifiable arcs, this would be easy: Let P(A) be its midpoint. But the problem is to find a P for all embedded arcs, rectifiable or not. Finally, (15+ years after first working on this) I think I've finally solved this problem. Hallelujah! Some other cases of the problem: b_n) S := all n-disks smoothly embedded in R^n I've recently solved this with a method that's much simpler than for case a). c_n) S := all smooth Riemannian metrics on the n-disk D^n. (In this case, maps into R^n above must be replaced with maps into D^n.) These problems have a curious status: n = 1: Just take the midpoint. n = 2: A bit tricky, but doable. n = 3: A non-constructive proof (based on a theorem of Allen Hatcher) shows a solution exists, but no explicit definition of the selection function P is known. n = 4: Unknown. n = 5: Unknown. n >= 6: No solution is possible. --Dan ________________________________________________________________________________________ It goes without saying that .