Consider some set X in R^{n} (say R^{2} 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, H_{wave}(X), is defined as

H_{wave}(X) = U_{r≥0}B_{r}(B_{r}(X) \ S) \ S,

_{r}(A) is defined as the union of all closed balls of radius r centered at points in A.

Theorem. For any X in R^{n}

H_{wave}(X) = X U int(H_{convex}(X)),

where int(A) denotes the interior set of A and H_{convex}(X) is the convex hull of X.

Corollary. cl(H_{convex}(X)) = ∩_{r}_{>0}H_{wave}(B_{r}(X)), where cl(A) denotes the topological closure of A.

We will prove the theorem and corollary in a later entry.

Ya he visto tu blog, muy interesante.

ReplyDeletePor cierto, el otro día estuve pensando en el teorema López-García de desgajamiento de yogures y se cumple también en las tres dimensiones(nº de cortes = nº total de elementos -1), ponte a ver si sacas la demostración del tema.

MOGTRO