
Re: an obvious geometric property (but proof required)
Posted:
Jun 12, 2014 9:21 PM


"quasi" <quasi@null.set> wrote in message news:g1gkp9letl24vp97fo3nuhvqimfpgu72ne@4ax.com... > David Hartley wrote: > >quasi wrote: > >> > >>(2) If S is a closed (but not necessarily simple) polygonal > >>curve, then V = R^2. > >> > >>James Waldby's very simple counterexample disproves (1). > >> > >>But (2) is still alive, and I'm convinced it's true. It's sort > >>of "obviously true", but I've been fooled before by flawed > >>visualizations, so even if it really is obviously true, that's > >>not good enough  proof is required. > > > >Just take scattered's example, with the lines slightly > >separated, and then join each extended line back to the end of > >the line it was originally joined too. As long as the > >extensions are long enough the new joining pieces are not > >"visible" from the centre. > > Nice. > > Ok, then I'll retreat to this: > > (2') If S is a simple closed polygonal curve (with finitely many > edges), then V contains all points of R^2 which are outside of S. >
This still doesn't work  we can make a sawtooth shape, "surrounding" a central point in the sense that all the point can "see" is saw teeth, each of which is partly obscured by another tooth. The teeth are mostly directly joined up to their adjacent teeth, but the last two of them join by a polygonal curve going right around the outside of all the teeth. This puts our central point on the outside of the polygon...
Regards, Mike.

