From: Martin Subject: Re: Problem with linear combinations of convex sets Date: 2 Jan 2000 13:24:41 -0600 Newsgroups: sci.math,sci.math.research,sci.math.symbolic Summary: Rusin goofs again Happy new year and thank you for your answer! But I think it is not correct. Dave Rusin wrote: > > In article <386A17DD.167EB0E7@rz.uni-karlsruhe.de>, > Martin wrote: > >Given $n-1$ convex compact subsets $K_i,\ i=1,\ldots,n-1$ > >of $\R^n$ with the property that > >$$\forall (x_1,\ldots,x_{n-1})\in K_1\times\ldots\times K_{n-1}: > >\{x_1,\ldots,x_{n-1}\} \mbox{ is linearly independent.} > >$$ > > > >Can I always find a fixed $x_n\in\R^n$ such that > >$$\forall (x_1,\ldots,x_{n-1})\in K_1\times\ldots\times K_{n-1}: > >\{x_1,\ldots,x_{n-1},x_n\}\mbox{ is linearly independent?} > >$$ > > > >For $n=2$ and $n=3$ I could prove that such an $x_n$ exists, > > Let's try n=4, then. I think I have a counterexample for the simplest case, > in which the K_i are just line segments. That situation can, except in > degenerate cases, be reduced to one like this: > K1 is the set of vectors [1,0,u,0] with |u| <= 1; > K2 is the set of vectors [0,1,0,v] with |v| <= 1; > K3 is the set of vectors [a,b,c,d]+t[e,f,g,h] with |t| <= 1. > There are some conditions to be satisfied by abcdefgh to guarantee linear > independence. I hope I have worked them out properly (I used techniques as > below); I think it's sufficient to have |g+-e| < |c+-a| and |h+-f| < |d+-b|. > Anyway, I chose a = 12, b = 7, c = 8, d = 4, e = 1, f = 5, g = 2, h = 3. > I'm afraid (or perhaps relieved because I would prefer a POSITIVE answer to my problem;-)) that these sets are not linearly independent. Take u=2/3, v=4/7, t=0. Then 12*(1,0,2/3,0)+7*(0,1,0,4/7)=(12,7,8,4). I think that the condition for linear independence is that not all 3x3 sub-determinants of |1 0 u 0 | |0 1 0 v | |a+te b+tf c+tg d+th| must vanish for the same value (u,v,t)\in [-1,1]^3, and this leads to the necessary and sufficient condition that the system u(a+te)=c+tg v(d+th)=b+tf has no solution in [-1,1]^3. (This yields under additional assumptions equations like |a+-e|<|c+-g|.) It would be interesting to know if under these conditions there exists a counterexample. But generally a counterexample with convex sets of positive n-dimensional measure should be easier to find than with straigth lines. Martin