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Topic: Lovelock and Rund: Star shaped set of points on a manifold
Replies: 7   Last Post: Jan 17, 2013 2:39 PM

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Shmuel (Seymour J.) Metz

Posts: 3,344
Registered: 12/4/04
Re: Lovelock and Rund: Star shaped set of points on a manifold
Posted: Jan 17, 2013 2:39 PM
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In <No-dnXx4EfewZ2vNnZ2dnUVZ_rGdnZ2d@megapath.net>, on 01/16/2013
at 02:06 PM, Hetware <hattons@speakyeasy.net> said:

>I'm having a bit of trouble grasping that concept. Let's take R^2,
>for example. I choose a point in the middle of my paper, and call
>it {0,0}. I now declare it to be a member of some open set U.
>The smallest possible U is some infinitesimal open disk centered
>on P.


What do you mean by "infinitesimal" and how is a disk with half the
radius not smaller?

>So am I correct in understanding that the "star shaped" set U is the
> infinitesimal disk centered on P, with parametrized curves
>radiating infinitesimally from P?


>Is any ball with x^i < 1 a permissible choice for U?

Not only that, but so is any other open ball containing P. However,
the definition of star shaped is more general; U need not be an open
ball, or even convex.

What definition does the book give?

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