Dan's closed-curve puzzle reminded me of a possible refinement to the idea of convexity. One property of convex plane figures is that a (straight) line cuts them into (at most) two pieces; while a (straight) line cuts a concave figure into various numbers of pieces, depending on the details of the figure & line position. Moreover, two straight lines cut a convex figure into at most four pieces, while concave figures can have more pieces. We could classify figures according to the spectrum of possible numbers of pieces from K cuts. Whether this is useful depends on what other properties we can connect to the classification. As a beginning, here's a conjecture: The intersection of two convex figures is convex. I think that the intersection of a concave figure (of cut-spectrum S) with a convex figure will have a cut-spectrum S' that's a "subset" of S. This is also related to the number and type of concavities for a figure. There may be theorems about the number of concavities for the intersection of two concave figures; maybe this number is limited to the sum of the concavity numbers of the two ingredients. Similar questions appear for the union of two figures; and for the pieces that result from cutting a concave figure. Rich ----------------- Quoting Dan Asimov <dasimov@earthlink.net>:
Let C be a C^oo simple closed curve. <snip>