Finite
Geometry Notes |

Steven H. Cullinane

Given a set X of points, certain families of subsets of X may have,

as

are concentric.) It may be that we can associate to each point of X

a subset of X, via an injection f: X → 2

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

we can write, in a cryptic but concise way, S = f(f(X)), and say that

f is an

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.