Historical preliminaries

-- Approximately a Subset

Let X be a finite set of N reals x_i : X = {x_1,x_2, .. ,x_i, .. ,x_N}
Let Y be a finite set of M reals y_j : Y = {y_1,y_2, .. ,y_j, .. ,y_M}

X is a subset of Y , X <= Y , if and only if

Sum(i=1)^(i=N) 1/N ( Product(j=1)^(j=M) |x_i - y_j| )^(1/M) = 0

Generalization and definition:

X is Approximately a subset of Y , X <= Y , if and only if

Sum(i=1)^(i=N) 1/N ( Product(j=1)^(j=M) |y_j - x_i| )^(1/M) = minimum

In words: The Arithmetic Mean of a Geometric Mean of distances should be minimized (with respect to suitable parameters).

Here the Geometric mean is defined according to WikiPedia .

So attention has been focussed upon Geometric Means for quite a while.

-- Continuous Geometric Means

-- Why doesn't MAPLE compute a definite integral ?

With accompanying executable & source code

Trying the same for a Circle has resulted in quite interesting threads. Here is the integral to be computed:

I = integral(t=0..2pi) ln( (a+cos(t))^2 + (b+sin(t))^2 ) dt

But what's really needed is: exp(I/(2.pi)) . (a,b) can be replaced by polar coordinates r, where r is the distance to the midpoint of the the circle divided by the radius of the circle. Anyway ..

Exact outcome for this integral ?

Weird simplification result with MAPLE

Interesting because of pattern matching with help of moments of inertia (which has not been entirely successful, BTW) is the

Determinant of Inertia

Rest of the story continued

here