Consider some set X in Rn (say R2 for simplicity of display) composed of several irregular components. Imagine now that this set X is made of some radioactive material so that it creates a radiation wavefront propagating from X outwards:
If seen from a sufficiently long distance, the radiation wavefront is indistinguishable from that of a convex set Y with the same exterior shape as X:
So, the convex hull of X (or something very similar to it, as we will see later) can be reconstructed from its radiation wavefront at some distance r by retropropagating the wavefront towards X. Let us see a graphical example where X consists of three points and the wavefront w is considered at a distance r larger than the diameter of X:
Retropropagating w is equivalent to considering the wavefront at distance r generated by the exterior of w:
The resulting shape includes X and approximates the convex hull of X (in this case, the triangle with vertices in X) as r grows. So, we have a characterization of convexity resorting only to our wavefront construction, which can be formalized by means of metric space balls:
Definition. Let S be some metric space and X a set of points in S. The wavefront hull of X, Hwave(X), is defined as
Hwave(X) = Ur≥0Br(Br(X) \ S) \ S,
Theorem. For any X in Rn
Hwave(X) = X U int(Hconvex(X)),
where int(A) denotes the interior set of A and Hconvex(X) is the convex hull of X.
Corollary. cl(Hconvex(X)) = ∩r>0Hwave(Br(X)), where cl(A) denotes the topological closure of A.
We will prove the theorem and corollary in a later entry.