And if the hull is convex, the shortest distance must be the edges around the hull. For otherwise, in the path ABCDA two edges, say AB and CD, cross at E; and then ACBDA, the hull, will be shorter than ABCDA, because AC+BD must be shorter than AEC+BED.
If the hull is not convex, neither will the shortest path be. So Sylvester's problem is the same as Raul's. Henry Rich Arie Groeneveld wrote: > For the mathematicians among you, have a look at: > http://tinyurl.com/5667g7 > > =@@i > > Raul Miller schreef: >> I am playing around with a problem, and I feel like I am missing >> a couple basic concepts, so I was wondering if anyone had >> any quick suggestions: >> >> If I have code which finds a shortest cycle which visits all points in a set: >> >> loopdist=: +/@:(+/&.:*:"1)@:(- 1&|.)"2 >> sortloops=:{.@(/: loopdist)@({~ (!@<: A.&i. ])@#)"2 >> >> And I use this to generate some random quadrilaterals >> (with coordinates of each chosen arbitrarily from some >> linearly distributed range): >> >> data=: sortloops (?.20 4 2$1000)+"1"_1(1000*4 5 #:i.20) >> >> What are the odds that any given quadrilateral is >> not concave? >> >> require'plot' >> plot j./"1 (,{.)"2 data >> >> Thanks, >> >> > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
