Does Everest have a topographic isolation?

Question

Here are three definitions of "topographic isolation".

Wikipedia

The topographic isolation of a summit is the minimum distance to a point of equal elevation, representing a radius of dominance in which the peak is the highest point.

wikidata.org

minimum distance to a point of higher elevation

gis.com

The topographic isolation of a summit is the minimum horizontal (great circle) distance to the nearest point of higher elevation.

The gis.com site has a table of "The 25 Most Topographically Isolated Mountain Peaks on Earth". At the top of this table is Mount Everest with an isolation of 40, 008 km. I looked up that 40, 008 km is about the circumference of the Earth.

But I am not sure this choice agrees with either of the definitions above because one of the criterion of the definition is not satisfied: there is no other point with equal (or higher) elevation for Everest (or at least the highest point of it anyway).

Then again, I did grab rather informal definitions from some wiki pages, so perhaps there are other definitions for more technical work that resolve this apparent contradiction.

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Miscellaneous afterthought: we should grant that we are talking about distances on a surface between peaks. There are taller mountains than Everest (e.g. Olympus Mons, but we are not considering the time-varying distance between planets in this question.

Answer 1

The topographic isolation of a summit is the minimum horizontal (great circle) distance to the nearest point of higher elevation.

If this definition is altered so that the distance is to the nearest point of equal or higher elevation, then the definition works. Starting at Everest, you would have to circle the Earth back to Everest before you find another peak of equal or higher elevation so the topographic isolation of Everset would be the circumference of the earth.

Answer 2

There is a contradiction, but you have choices in deciding how to resolve it. Here are two:

1. Everest doesn't have a topographic isolation; call it undefined. It can be as simple as that, but maybe ignores our intuition that Everest is the most isolated.
2. Find a more general definition that preserves the topographic isolations as they are already calculated. One approach to doing this is to calculate topographic isolation as normal for any peak that isn't the highest peak in the set. Then for the highest peak in the set take its topographical isolation to be the greatest distance from itself to anywhere in the space. On a sphere speckled with mountains, this will be the circumference of the sphere (i.e. a great circle's length).

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While not immediately useful to the original question, this notion of topographic isolation can be readily generalized to many mathematical surfaces embedded in finite dimension and equipped with a suitable metric. For surfaces over an infinite support, the "topographic" isolation of a global maximum may be unbounded (or informally, "infinite").