[seqfan] Re: partition of a circle

[Ed Jeffery]

Neil,

Sorry to butt in again, but I think Valette-Z. mean that the pieces have
null intersection except at their boundaries, with their disjoint union
being the entire disk (circle); i.e., as with tiles, the pieces must fit
together without gaps or overlaps.

As for convexity, surely the pieces must be convex, since otherwise the
example (for n = 3) I sent to you would be a valid candidate. To describe
that example again: draw a smaller circle inside the larger circle to be
divided, then draw two lines, each extending from the smaller circle to the
larger one, and which divide the annulus into two pieces without either
line intersecting the interior of the smaller circle or disk. The two
pieces forming the annulus can't be convex.

Ed

> Wolfdieter, Yes, I agree that we need more conditions
> that will rule out your example.

> In fact in the Valette-Z. paper, they add the condition
> that the partition must be admissible, which
> they define to mean that if K and L are two distinct pieces,
> then the intersection (boundary of K) intersect (boundary of L)
> is connected.
> This rules out your example, since the two big pieces has intersection
> which is not connected.

> By the way, they do NOT require that the pieces be convex. (That would be
> another way to
> rule out your example)

> Neil