Question (repeated here for convenience):
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Assume a receiving antenna is in the center of a sphere and the
received signal is coming in equal amounts from all points on the
surface of the sphere. Which receiving antenna would capture more
power, an omni or a high gain beam? There are no noise and no losses.
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First, thanks for all the comments. They have helped me better
understand the answer. I am leaning toward the belief that the omni
(isotropic) antenna would capture more power and, as odd as it may
seem, would have more gain than a high gain beam (or any other
directional antenna for that matter). Here is my thinking:

This is a very unusual RF field. Usually the field is assumed to be
planar with coherent rays - then antennas behave as expected. But this
field originates uniformly from all points on the surface of a sphere.
It does not spread but converges at the focal point of the sphere.

An isotropic antenna placed at the focal point would collect all of
the rays whereas a directional antenna at would not.

Therefore, in this particular situation, the isotropic would have
higher gain and capture more power than any directional antenna.

Please correct me if I am wrong.

Ron, W4TQT