Re: What is the mathematical definition of round?

On Sun, 12 Oct 2008 18:38:34 EDT, T.H. Ray wrote:
Dave Seaman wrote

On Sun, 12 Oct 2008 05:27:01 EDT, T.H. Ray wrote:
william elliot wrote

On Sat, 11 Oct 2008, Herman Rubin wrote:
Dave <dave_and_darla@xxxxxxxx> wrote:
On Oct 10, 8:56=A0pm, BURT
<macromi...@xxxxxxxxx>
wrote:

What makes something geometrically round?

In the geometrical sense, round means "having a
circular shape." So
something is round if and only if it is a
circle.

No, it is much less. For closed convex sets,
round
means
that every point is an extreme point.

Every point of the set is an extreme point or
every point of the boundary is an extreme point?

Herman Rubin specified closed (i.e., compact) and
convex
set. His definition is sufficient. That every
point
lies extreme to a fixed point guarantees
continuous,
uniform curvature on the boundary.

What does "extreme to a fixed point" mean? I know
what an extreme point
of a convex set is, but by that definition, only the
boundary points of a
disk qualify as extreme points.

A disk is "round," is it not? The definition can
be extended to the underlying disk manifold of higher
dimensional objects.

We agree that a disk is round, but you didn't answer my question. What
does "extreme to a fixed point" mean?

It appears to me that a workable definition is that a compact convex set
is round if all of its boundary points are extreme points, as William
Elliott suggested. According to this definition a disk is round, but a
circle is not. You seemed to take issue with that suggested definition.
I am merely asking for an explanation of your objection.

A circle is ruled
out because it is not
convex.

And a circle is not "round" in any but the colloquial
sense. A closed curve is not necessarily round.

See above. A circle is not round by the proposed definition because it
is not convex. The question remains, why are you convinced that a disk
is round, despite the fact that not all of its points are extreme points?
Or have I misunderstood your claim? It's hard to tell, because you have
not explained what "extreme to a fixed point" means.

Moreover, a convex set having an elliptical
boundary also has
the property that every boundary point is an extreme
point, even though
the curvature is not uniform.


Then maybe "uniformly extreme" is the qualifier.

What does "uniformly extreme" mean?


--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
.

Relevant Pages

  • Re: What is the mathematical definition of round?
    ... For closed convex sets, round means that ... every point of the boundary is an extreme ... You've added compact. ... If a "round" set is defined to be a nonempty, compact, convex set, all of ...
    (sci.math)
  • Re: What is the mathematical definition of round?
    ... is round if all of its boundary points are extreme points, ... which is the same as the boundary of the included disk. ... then the extreme points are always ... not topology. ...
    (sci.math)
  • Re: Integer lattice
    ... All such functions f form a locally compact convex cone. ... to show that there is only one extreme point. ... S is weak-star closed and that the intersection Q ... weak-star compact convex set K (using the ...
    (sci.math)
  • Re: Integer lattice
    ... All such functions f form a locally compact convex cone. ... to show that there is only one extreme point. ... S is weak-star closed and that the intersection Q ... weak-star compact convex set K (using the ...
    (sci.math)
  • Re: Integer lattice
    ... All such functions f form a locally compact convex cone. ... to show that there is only one extreme point. ... S is weak-star closed and that the intersection Q ... weak-star compact convex set K (using the ...
    (sci.math)
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