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From: cbrown on 13 Aug 2010 00:03
On Aug 11, 7:01 am, Kent Holing <K...(a)statoil.com> wrote:
> By drawing a circle on a map, can I always claim that there are at least two diametrically opposite points on the circle with the same height above the sea level?

Others have given solutions; I add that this appears to be a special
case (for n = 1) of the Borsuk-Ulam theorem (Wikipedia has an article)
which states that given a continuous function f from an n-sphere into
Euclidean n-space, f maps some pair of antipodal points to the same
point. (Two points on an n-sphere are called antipodal if they are in
exactly opposite directions from the sphere's center.)

Of course, in practice, the function f in your is not strictly
speaking continuous: the circle may pass over an area which "leans
out" (rock climbers call this an overhang; but you can image the
circle passing through an inverted cone of rock). And depending on how
detailed we imagine the definition of 'height' might be, a nit-picker
might note that such discontinuities are caused by tiny crystals of
stone causing similar discontinuities on the sub-millimeter scale.

But if we ignore such very local phenomena, then 'height above sea
level' can be thought of as being continuous, and then the answer to
your question would be 'yes'.

Cheers - Chas
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