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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