AFAIR disk is not a mathematically defined term for high dimensions. Ball is as you defined it, where one can argue whether the distance is "strictly less" or " at most" r. Sphere is the definition where the distance is equal to r.
R.E. Boss > -----Original Message----- > From: Programming [mailto:[email protected]] > On Behalf Of Murray Eisenberg > Sent: zondag 20 augustus 2017 16:33 > To: [email protected] > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > > 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 > > > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
