Possibly I had an incomplete description of the problem but the use of
"point" brought up the problem of "how big is a point?".
Your definitions below do help -and also make the problem more
interesting for n>3
Don
On 2017-08-20 7:33 AM, Murray Eisenberg wrote:
Re Don Kelly’s comment:
NO the ball of radius r at a point p in n-space is the set of all points of
distance strictly less than than r from p;
the disk of radius r at a point p in n-space is the set of all points
of distance at most r from p.
“Infinitesimal” has utterly nothing to do with it, nor does transfinite math (although the set of points in such a ball, or such a disk, is definitely infinite and, in fact, uncountable.
Re Jimmy Gauvin’s comment:
It is utterly trivial to prove that a ball (or disk) in euclidean n-space
is convex. It requires nothing more than what is commonly taught in standard
courses in sophomore linear algebra today. Specifically, basic properties of
the euclidean norm, including the triangle inequality.
On 2Sat, 19 Aug 2017 18:03:49 -0700,Don Kelly <[email protected]
<mailto:[email protected]>> wrote:
If one considers a point as infinitesimal -as usually considered-, then
we have an infinite number of points at an infinitesimal distance from
the origin and at a larger distance from the origin there are still an
infinite number of points on the surface and an infinite number of
points enclosed . Isn't this getting into transfinite math?
What's the point?
Don Kelly
On 2017-08-15 8:23 PM, Jimmy Gauvin wrote:
The construction of the sphere implies it cannot be convex but you will
have to find a topologist to prove it to you.
The sphere is the collection of points whose distance to the origin is
equal to the radius of the sphere.
The ball or volume is comprised of the points whose distance to the origin
is equal or smaller than the radius of the sphere.
——
Murray Eisenberg [email protected]
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 240 246-7240 (H)
University of Massachusetts
710 North Pleasant Street
Amherst, MA 01003-9305
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