Then make it a linear programming problem: Assign non-negative weights Wi to vectors Vi, with sum Wi = 1. Put W = sum WiVi. Then require the dot products W.Vi >=0. The dot products are linear in the weights, so it's a conventional linear programming problem. If it's feasible, you have a solution W. We need to prove that if there's a solution, there's one in the convex hull of the Vi. This seems plausible, but I don't see an immediate proof. Rich ------ Quoting Eugene Salamin <gene_salamin@yahoo.com>:
Not quite. Put a bunch of atoms at Lat 89 deg N, Long 0 deg. Put one atom atLat 89 deg S, Long 0 deg. These all fit within the hemisphere bisected by the Greenwich Meridian. But W is near 89N so its dot product with that last atom is negative.
-- Gene
________________________________ From: "rcs@xmission.com" <rcs@xmission.com> To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Sent: Wednesday, September 18, 2013 5:04 PM Subject: Re: [math-fun] Probability that all molecules of a gas are in one half of the container
Wrt part 2 of the puzzle, to provide a test, the following might work: If Vi are the locations of the atoms, compute W = sum Vi. Then see if all the dot products W.Vi are positive.
Rich
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