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I have just noticed that the mathworld article has nice additional information. It should be incorporated here. -- GaborPete ( talk) 06:38, 26 January 2010 (UTC) reply

OK, I have done that. In fact, it's a slightly different notion of conformal radius, also called logarithmic capacity. Now proper connections with the article capacity of a set should be established. -- GaborPete ( talk) 08:30, 8 February 2010 (UTC) reply

"When D ⊂ C is a connected, simply connected compact set, then its complement E = D^c is a connected, simply connected domain in the Riemann sphere that contains ∞"

This can't quite be right. We need some kind of local path-connectedness, neighborhood retract, or other mild regularity property. Otherwise you can make a loop with the topologist's sine curve that is simply-connected according to the definition, but whose complement is not connected.

2001:171B:2274:7C21:60E8:E9FE:B633:C9CE ( talk) 12:22, 5 March 2022 (UTC) reply