convex functions on L_1 spaces

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Article #49

#49

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

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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.

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Path: arc1-ca.pugleaf.net!sparky!uunet!europa.asd.contel.com!darwin.sura.net!wupost!sdd.hp.com!ux1.cso.uiuc.edu!news.cso.uiuc.edu!usenet
From: fernand@acf9.nyu.edu (Chris Fernandes)
Subject: convex functions on L_1 spaces
Nntp-Posting-Host: acf9.nyu.edu
Message-ID: <fernand.715217944@acf9>
Summary: Do strictly convex functions on closed convex and bounded set in L_1 achieve Minimum?
Sender: Daniel Grayson <dan@math.uiuc.edu>
X-Submissions-To: sci-math-research@uiuc.edu
Organization: New York University
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Approved: Daniel Grayson <dan@math.uiuc.edu>
Date: Sun, 30 Aug 1992 23:39:04 GMT
Keywords: convex, L_1
Lines: 19