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Topic: Generalized Cauchy-Schwarz
Replies: 7   Last Post: Apr 13, 2012 3:09 PM

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Kaba

Posts: 289
Registered: 5/23/11
Re: Generalized Cauchy-Schwarz
Posted: Apr 11, 2012 5:57 PM
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Kaba wrote:
> Kaba wrote:
> > Therefore, sharpening the problem, the problem is to characterize when
> > triangle inequality holds if B is indefinite.

>
> Let's not forget:
>
> Triangle inequality holds for f
> <=>
> The sublevel sets of f are convex (f is quasiconvex).
> <=>
> The f is convex.
>
> Here quasi-convexity happens to be equivalent to convexity although I am
> not sure why (possibly because of homogenuity of f).


Scrap that. Quasiconvexity does not imply triangle inequality or
convexity here. The following hold:

Triangle inequality holds for f
<=>
The f is convex

and

The f is convex
==>
The sublevel sets of f are convex (f is quasiconvex).

--
http://kaba.hilvi.org



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