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Re: [Phys-L] rms / conic / arithmetic / geometric averages

Suppose we generalize JD's 'conic' average for a truncated cone to a truncated hyper-cone in d-dimensions. What I have in mind here is a linearly tapered object capped at each of its 2 ends by bases which are (d-1)-dimensional hyper-spherical balls of radius r_1 & r_2 respectively. In this case the corresponding 'hyper-conic' average of those base end caps needed for evaluating the corresponding hyper-volume of the corresponding hyper-cylinder is
R_avg = (((r_2^d - r_1^d)/(r_2 - r_1))/d)^(1/(d-1)) .
The quotient of differences here always results in a finite sum whenever the dimensionality, d, is a integer of at least 2. In this case the various terms in the sum for the average are products of powers of r_1 & r_2 such that the sum of the powers is d - 1, & the sum has d terms. For example when d = 4 we have
R_avg =((r_2^3 + (r_2^2)*r_1 + r_2*(r_1^2) + r_1^3)/4)^(1/3).
When d = 3 we have JD's conic average, i.e. R_avg = ((r_2^2 + r_2*r_1 + r_1^2)/3)^(1/2).
And when d=2 we have the ordinary arithmetic average.

David Bowman