another way to generalize convexity is to play with the quantifiers. S is convex if: for every x,y in S, the line segment L(x,y) between x and y is in S. what does it mean if: for every x,y,z in S, at least one of the line segments L(x,y), L (x,z), L(y,z) is in S? (some authors call such sets 3-convex, some call them Valentine convex) more generally, for any n points, if at least m of the (n choose 2) lines are in S? are there non-trivial examples of such sets? what are their properties? erich On Jan 10, 2010, at 9:20 PM, rcs@xmission.com wrote:
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>
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