We classify all subsets
S of the projective Hilbert space with the following property: for every point
, the spherical projection of
to the hyperplane orthogonal to
is isometric to
. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes
with the property that for every
... [Show full abstract] the conditional distribution of given that coincides with the distribution of for some function . A basic example of such process is the stationary zero-mean Gaussian process with covariance function . We show that, in general, the process Z can be decomposed into a union of mutually independent processes of two types: (i) processes of the form , with , , and (ii) certain exceptional Gaussian processes defined on four-point index sets. The above problem is reduced to the classification of metric spaces in which in every triangle the largest side equals the sum of the remaining two sides.