convexity and infinite convex combinations
Roland Strausz
I am looking for references concerning infinite convex combinations and
would like to know if a converging convex combination of infinitely many
points lies in the convex hull of these points. I believe to have a
proof for the specific setting I am considering, but it is rather
cumbersome and hardly elegant. I looked through several books on
convexity, but the topic of infinite convex combinations is never really
touched. It would be most helpful, if somebody out here could help me
further. I need the result for non-countable many elements. Below is
more information concerning my problem:
To be more specific, let's first consider the countable case: Let
A={a_1,a_2,...} be a subset of R^n (n finite), containing infinite, but
countable many elements. Let the scalars q_1,q_2,... be such that
q_i>0 for all i=1,2,... and \sum q_i=1. Also let \sum a_i q_i converge
to a point w.
Claim: The point w lies in the convex hull of A.
The claim sounds reasonable to me, but I haven't found it anywhere. I
am wondering why not. If the claim is not generally true, are there any
conditions on A for the result to hold? (Closedness would of course do,
but in my setting A would not necessarily be closed. Boundedness would
be ok, though.)
Now I even want to take it one step further and do it for non-countable
many elements. Consider the measure space (R^n,B(R^n),q) with B(R^n)
the Borel sets and q a probability measure. Now consider a measureable
function f:R^n->R^m and define the point set A={f(x)|x \in R^n}.
Question: If the integral \int f dq converges to a point w, does w lie
in the convex hull of A?
Also here boundedness of A is ok, but A is not necessarily closed.
dear regards,
Roland
Herman Rubin
>I am looking for references concerning infinite convex combinations and
>would like to know if a converging convex combination of infinitely many
>points lies in the convex hull of these points. I believe to have a
>proof for the specific setting I am considering, but it is rather
>cumbersome and hardly elegant. I looked through several books on
>convexity, but the topic of infinite convex combinations is never really
>touched. It would be most helpful, if somebody out here could help me
>further. I need the result for non-countable many elements. Below is
>more information concerning my problem:
This is true in finite dimensional spaces. I believe the first
published proof of this was by Wesler and myself in 1958. One
way of looking at the problem is that the intersection of hyperplanes
of arbitrary dimension which have probability one is a hyperplane
with this property. This can fail in infinite dimensional spaces.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Robert Israel
|> To be more specific, let's first consider the countable case: Let
|> A={a_1,a_2,...} be a subset of R^n (n finite), containing infinite, but
|> countable many elements. Let the scalars q_1,q_2,... be such that
|> q_i>0 for all i=1,2,... and \sum q_i=1. Also let \sum a_i q_i converge
|> to a point w.
|> Claim: The point w lies in the convex hull of A.
True. Let U be the interior of the convex hull of A. We may assume U is
nonempty (otherwise restrict everything to an affine set containing A).
If w is not in U, we can separate it from U by a hyperplane, i.e. there
exist v in R^n such that v.u < v.w for all u in U. In particular,
v.a <= v.w for all a in A. But sum q_i v.(a_i - w) = 0 with all q_i > 0,
so all v.a_i = v.w, and thus v.u = v.w for any u in the convex hull
of A, contradiction.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2