rcs at xmission.com rcs at xmission.com 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.

--seems to me, rcs here (after removing crud) has a perfectly good reformulation of the problem: there is an empty hemiball if and only if the linear program x . V[i] < 0 for each i=1,2,3...,N has a solution-vector x. There is no need to have a solution "within the convex hull of the V[i]" nor to make weighted combinations of the V[i]. The solution to the gas problem thus also tells us the probability that an LP of this form has a solution (when the V[i] random on sphere). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)