Let A_{0}=a and B_{0}=b, and for all natural i let A_{i} be the arithmetic mean of A_{i-1} and B_{i-1}, and let B_{i} be the harmonic mean of A_{i-1} and B_{i-1}.
If there is a limit to the A_{i} and the B_{i} sequences, and if the limit is the same for both sequences, this limit is the geometric mean of a and b.
To see that this coincides with the standard definitions on the positive reals, consider that the product of the means is ab, so for any i, A_{i}B_{i}=ab.
Over the reals the two sequences clearly converge, and clearly do so to a single value (consider that the distance between A_{i} and B_{i} shrinks by a factor of two at least in every iteration, and that these values bound the values for all future iterations), so the limit must be some x for which x^{2}=ab, hence it is the geometric mean.
The tools used to define the geometric mean in this way clearly do not require anything beyond what was assumed in the question. We note that we used the fact that S is a metric space not only in the definition of the arithmetic mean but also in our ability to describe a limit to a sequence in S. However, we did not use at any point the property that x^{-1} is an involution. Any function would have done.
As we have only used the equality (A+B)/2 * 2/(A^{-1}+B^{-1}) = AB, the result is applicable to any field. However, not in every field is convergence guaranteed, nor is it true in all fields where convergence is guaranteed that the proof of this fact is easy to see. I leave to the readers the example of complex numbers in the first and fourth quadrant as an example where convergence is guaranteed and proving this is not difficult. The fact that convergence is guaranteed in the (non-field) of positive definite matrices is far more difficult. There, the limit is known as the matrix geometric mean, and can be written explicitly as
This expression is symmetric in A and B, and if AB=BA, then A#B=(AB)^{1/2}.
All of this month's solvers reached this one solution, except for Mengxiao Zhang who deserves a special mention for generalising the geometric mean in another way. In Mengxiao's solution, the geometric mean is the element x in S satisfying
if such an element exists and is unique.
Credits for this month's riddle begin with Vitali Milman, who not only introduced me to this problem to begin with but also dug (helped by Liran Rotem) amazing amounts of background information about it and its origins.
The redefinition of geometric means on numbers is "folklore" and its exact origins we have not been able to ascertain. The geometric mean on positive definite matrices was initially considered in the paper
W. Pusz and S. L. Woronowicz. "Functional calculus for sesquilinear forms and the purification map". Reports on Mathematical Physics, 8(2):159-170, 1975
and then in book form in
T. Ando, Topics on Operator Inequalities, Lecture Notes Hokkaido Univ., Sapporo, 1978.
The first appearance of the definition described here, detailing the iterative process that converges to the geometric mean and proving convergence, is
F. Kubo and T. Ando, "Means of positive linear operators", Math. Ann. 246 (1980) 205-224.
Geometric means for matrices have since been studied extensively. A detailed review of various possible definitions for it can be found here:
J. Lawson and Y. Lim. "The Geometric Mean, Matrices, Metrics, and More". The American Mathematical Monthly, 108(9):797-812, 2001.
Vitali and Liran's interests, specifically, are around the use of this definition in the context of finding a geometric mean between two convex bodies. They were the first to employ these definitions to define this geometric mean. Their results are as yet unpublished. They can be cited as
V. Milman and L. Rotem. "Non-standard constructions in convex geometry: Geometric means of convex bodies". Submitted.
L. Rotem. "Algebraically inspired results on convex functions and bodies". Communications in Contemporary Mathematics, To appear.
V. Milman and L. Rotem. "Geometric means of convex sets and functions and related problems (part of Convex Geometry and its Applications)", Oberwolfach Reports 56/2015, To appear.
Although unpublished, Liran has graciously made the first of the accepted papers available electronically here, on his home page. The second accepted paper can be found on page 51, here.
Two earlier precursors of this redefinition appear in
E. Asplund. "Averaged norms". Israel Journal of Mathematics, 5(4):227-233, 1967
and
V. P. Fedotov. "Geometric mean of convex sets". Journal of Soviet Mathematics, 10(3):488-491, 1978.
The former does not use the terminology of geometric means, the latter uses the terminology, but does not use this type of construction.
Regarding the latter, Vitali related to me the following amazing history, which I am copying here:
I first read Asplund's paper in 1969. He considered ||x||^{2}. It was very important to have 2-homogeneity. So, for two equivalent norms, one uniformly convex and another uniformly smooth, he built a third equivalent norm which had both these properties. This is what I now call the "geometric means" of the two original norms. |
For duality he used the Legendre transform, and this was the reason 2-homogeneity was important. (The Legendre transform preserves 2-homogeneity.) |
I liked his construction very much and thought about it that time a lot, and transferred it to many other situations, and particularly to convex bodies. |
However, the big theorem I wanted to prove with this did not crystallise and I did not publish many of the things I did. (Thirty years later a counter-example was built and this was a great development.) However, in 70-71 I visited Leningrad and gave a few talks. In particular, in one functional analysis seminar by Vulich I talked about this and many other things that made me interested in this construction. It is my understanding that a PhD student, Fedotov, was present. (I later learned that he worked on this seminar.) |
So, after I emigrated to Israel (in 1973), he published a paper with a few additional remarks to what I told in the seminar. I understand why in his Note there are no mentions of my talk and my name. At the time it was absolutely forbidden to mention any name of an emigrant from Russia in publications. So, most probably, he wrote it, but it was taken out. |
I definitely know this happened with another paper which was influenced by that talk of mine, by B. Tsirelson. He also was a PhD student at that time. He was in the seminar and was very excited by some questions I asked, and wrote a paper which became the most important paper in infinite dimensional Banach Space theory for decades. (The Fields medal of Gowers, for example, is also a development and continuation of that paper.) This was the first paper by Tsirelson. (He is a great mathematician; he turned to Probability after this.) So, he sent me his preprint before I left the Soviet Union (a month before) and it had many references to my talk. However, when it was published not one reference remained. When we met 20 years later, I did not ask him about it (I knew this rule), but he came to me himself to tell how he was forced to take them out and how it was done. |
Once again, many many thanks to Vitali and to Liran for all their help in chasing up the origins of this riddle and for introducing me to it to begin with.