The ex-sphere is tangent to the base facet externally, and to the other facets internally; -r_EX denotes its radius. Here's a succinct proof, in the spirit of Klamkin (and Bogdanov). Denote n-space simplex interior content (volume) by V , boundary content (surface area) by U . Then (n V)/r_EX + (n V)/r_IN = 2 F_0 - U + U = 2 F_0 = (n V)2/h ; now cancel n V . QED On 2/20/16, Dan Asimov <dasimov@earthlink.net> wrote:
I would appreciate a much better reminder:
How is r_EX defined?
—Dan
On Feb 19, 2016, at 6:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reminder --- r_EX is negative!
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
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