convex functions on L_1 spaces

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2 total messages Started by Chris Fernandes Sun, 30 Aug 1992 23:39

#49

Author: Chris Fernandes
Date: Sun, 30 Aug 1992 23:39

18 lines
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I have a question regarding convex functions on L_1 spaces.

Consider the space L_1.
Let U: L_1->R  be a strictly convex function.
Let G be a closed convex and bounded set in L_1.

Is it then true that the function U achieves its minimum on the set G?

I have not been able to find this result in any reference book.  Intuitively,
 the result seems to be true, but I cannot come up with a clean proof.

I do know that the set G is compact in the weak* topology, but on L_1
the weak* and the weak do not coincide.  For example, in L_2, the set
is weakly compact and thus the minimum is achieved.

Hopefully  yours,

Indrajit, mailing from friend's account.

Re: convex functions on L_1 spaces

#51

Author: Robert B. Israel
Date: Mon, 31 Aug 1992 17:00

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In <fernand.715217944@acf9> Indrajit, using the account of 
   fernand@acf9.nyu.edu (Chris Fernandes) writes:

>I have a question regarding convex functions on L_1 spaces.

>Consider the space L_1.
>Let U: L_1->R  be a strictly convex function.
>Let G be a closed convex and bounded set in L_1.

>Is it then true that the function U achieves its minimum on the set G?

Not true in l_1: let G = { x: x_i >= 0, sum_i x_i = 1 } and U(x) = sum_i x_i^2.
The infimum is 0 (consider U(x) where x_i = 1/n for 1 <= i <= n, 0
otherwise), but it is not achieved.  I don't have an example for other
L_1 spaces, but it probably wouldn't be hard to construct.

>I have not been able to find this result in any reference book.  Intuitively,
> the result seems to be true, but I cannot come up with a clean proof.

>I do know that the set G is compact in the weak* topology, but on L_1
>the weak* and the weak do not coincide.

What weak* topology?  L_1 spaces (except for l_1) are generally not dual
spaces.

>  For example, in L_2, the set
>is weakly compact and thus the minimum is achieved.

>Hopefully  yours,

>Indrajit, mailing from friend's account.

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4

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