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Topic: curvature and convexity
Replies: 1   Last Post: Apr 13, 2011 9:36 AM

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Dan Luecking

Posts: 26
Registered: 11/12/08
Re: curvature and convexity
Posted: Apr 13, 2011 9:36 AM
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On Tue, 12 Apr 2011 06:50:58 -0600, d <cliffordsmyth@gmail.com> wrote:


For the differential geometers out there:  "Given B, a compact subset
of R^n with non-empty interior is the condition that B is strictly
convex equivalent to some other condition involving some sort of
curvature on S the boundary of B?"

Depends on the meaning of "some sort" and "strictly convex".

I understand strictly convex to mean that a chord connecting
any two points on the boundary has only its endpoints on the
boundary and the rest in the interior of the set.

Consider the function:
      f(x) = \sum 2^{-n}|x-r_n|
where {r_n : n=1,2,3,...} is an enumeration of the
rationals. Then the supergraph {(x,y): y >= f(x)} is
strictly convex, but f''(x)=0 almost everywhere.

However, f'(defined at every irrational) is strictly
increasing, so "some sort" of curvature condition might
be said to be satisfied, defined in terms of the turning
of tangent lines.

Caveat: I am a geometer, but not a differential geometer.


Dan
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