Relatively Chubby Squirrels

Everything is bigger in America – even the squirrels. This one lives with its fellow scholars on the Diag.

Juha liked my suggestion to call our sets “relatively chubby”. So don't you laugh at my naming attempts there in Helsinki. I've got authority to back me up now. (Well, strictly speaking Juha only commented on the Finnish version “suhteellisesti pullea” (tuttavallisesti tietenkin “suhteellisen pullea”).)

An open bounded set U in a metric space is relatively chubby if the following holds for some constants R>0 and c>0: For every ball B(x,r), r<R, with distance (proportional to) r from U one can find a ball B(y,cr) inside U with distance (proportional to) r from B(x,r). In other words, if there is a lot of room just outside of U, there should also be room inside U at that location, but on the other hand if there isn't much room outside U, U can be thin as well. Accordingly, U is relatively chubby.

(What we really want to say above is: For every sufficiently small Whitney cube Q in the complement of the closure of U there exists a proportional Whitney cube inside U and close to Q.)

Currently I'm debating whether this really is the right concept or not.

More squirrels here.