Is there a good approximating polygon for every smooth set? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:43:30Z http://mathoverflow.net/feeds/question/58638 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58638/is-there-a-good-approximating-polygon-for-every-smooth-set Is there a good approximating polygon for every smooth set? domotorp 2011-03-16T14:01:01Z 2011-03-16T14:48:13Z <p>Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is a (1+epsilon) scaled copy of P that contains S?</p> http://mathoverflow.net/questions/58638/is-there-a-good-approximating-polygon-for-every-smooth-set/58642#58642 Answer by Tapio Rajala for Is there a good approximating polygon for every smooth set? Tapio Rajala 2011-03-16T14:48:13Z 2011-03-16T14:48:13Z <p>I do not see how scaling could give such a property. Consider an annulus in the plane (remove a part of it to make it a Jordan domain and also make it smooth if you want). Scaling any polygon inside the annulus by $1 + \epsilon$ makes also the "missing ball" inside the annulus larger.</p>