averages distances

Since the results and surviving conjectures from the 2 threads

"averages distances to a compact set"

and

"average distances to a finite set"

are essentially the same, it seems desirable to unify the discussion
into one thread.

Here is a new conjecture, adapted from the previous ones,
incorporating some of the results established so far ...

Fix a positive integer n > 1.

Let S_finite be the set of nonempty finite subsets R^n such that F has
at least 3 points which are not collinear.

Let S_continuous be the set of nonempty compact subsets of R^n which
are the closures of their interiors.

For the remainder of the discussion, first choose one of the above
versions of S, either finite or continuous.

For the finite version, let S = S_finite.

For the continuous version, let S = S_continuous.

Assume now that the version has been specified.

Fix a positive real k.

For x in R^n, Y in S, let

a(x,Y) = avg({d^k(x,y) | y in Y})

where d is the ordinary Euclidean distance function, and d^k denotes
the k'th power of d.

Thus, each Y in S induces a function a_Y : R^n --> R, defined by

a_Y(x) = a(x,Y)

Conjecture:

For any Y in S, there is a unique point p in R^n and a positive real
number b such that

a_Y(x) = g_k(d(x,p)) + b

where g_k is a nonnegative, continuous, increasing, bijective function
from [0,infinity) to [0,infinity), depending on k, but independent of
Y.

Corollaries:

Assuming the truth of the above conjecture, we get some easy
corollaries:

(1) The function a_Y has absolute minimum value b, achieved only at x
= p.

(2) For all c > b, the set {x | a_Y(x) < c} is an open ball, centered
at p.

(3) If Y1, Y2 in S have the same p (the same "center"), then a_Y1 =
a_Y2. In particular, knowledge of the function a_Y gives no
information about Y, except for the location of p.

Remarks:

Using the ideas suggested in the previous threads by Mariano
Suárez-Alvarez, it's not hard to prove the conjecture for k = 2.

I think such a proof can perhaps be generalized to work for any even
positive integer k. Odd positive integer values of k seem more
problematic, but may be doable with similar reasoning. As far as
arbitrary positive real values of k, I'm not sure.

quasi
.

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