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Steven H. Cullinane    Inscapes II.   Query.   September 22, 1982.

Given a set X of points, certain families of subsets of X may have,
as families, some property s.  (Example:  the families of spheres that
are concentric.)  It may be that we can associate to each point of X
a subset of X, via an injection f: X → 2X ,  in such a way that the
f-image, in turn, of this subset of X (i.e., the family of f-images of its
points) is in fact one of the families of subsets of X that have property s.

If the map f gives rise in this way to the set S of all such s-families,
we can write, in a cryptic but concise way, S = f(f(X)), and say that
f is an inscape of S.

Query:  What known results can be stated, after the appropriate definition
of S, in the form "There exists an inscape of S"?

Addendum of Oct. 10, 1982.  A more precise definition:

 Let X be a non-empty set. Let P(X) denote the set of all subsets of X. Let S ⊂ P(P(X)). Suppose there exists an injection f: X → P(X) such that, for any σ ∈ P(P(X), σ ∈ S if and only if ∃ x ∈ X such that σ = f(f(x)) = {f(y)| y ∈ f(x)}. Then f is an inscape of S.

This notion arises naturally in studying the action of a symplectic polarity in
a projective space.  One of course wonders whether it has arisen previously
in any other context.