On Thu, Dec 22, 2011 at 2:26 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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] -> P,
This requires not just a "family" S, but a "topological space" S. In the cases below, you specify the set S, but not the intended topology on S. It's not at all clear to me what the intended topology is on, say, the set of embedded arcs in the plane. Are you considering them just as subsets of the plane, or as parameterized arcs, so that two parameterizations of the same arc are considered different points of S? Assuming the latter for the moment, at least two topologies occur to me that seem natural; the topology of pointwise convergence and the topology of uniform convergence. Which is intended in the problem?
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:
I'm confused; a smooth Riemannian metric on D^n is not a subset of D^n. So I don't know how to make sense of the requirement that P(s) is a member of s. P(s) is a point in D^n; S is a smooth Riemannian metric on D^n.