When is the area of the convex hull of a tree-like linkage maximal? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:56:41Z http://mathoverflow.net/feeds/question/23056 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23056/when-is-the-area-of-the-convex-hull-of-a-tree-like-linkage-maximal When is the area of the convex hull of a tree-like linkage maximal? Gjergji Zaimi 2010-04-30T00:05:23Z 2010-04-30T00:05:23Z <p>This is inspired from <a href="http://mathoverflow.net/questions/22767/is-the-area-of-a-polygonal-linkage-maximized-by-having-all-vertices-on-a-circle" rel="nofollow">this</a> recent question. Given in the plane a tree-linkage (fixed length rigid edges, vertices are flexible joints, connected and no cycles) is there a simple description of when the area of its convex hull is maximal? Or alternatively an algorithm to find this maximal configuration? I think I checked the following simple cases:</p> <p>In the case when we have three edges connected at a point (say the edges are \$OA,OB,OC\$). Then the maximal area is achieved when \$O\$ is the orthocenter of \$ABC\$. </p> <p>In the case when we have a free path \$A_1,A_2,\dots,A_n\$ the maximal area of the convex hull is attained when all the vertices lie on the circle with diameter \$A_1A_n\$.</p>