I wrote: << In the plane: Let C be a C^oo simple closed curve. Let a "double-normal" be a line segment whose endpoints lie on C and which is normal to C at each of them. C must have a double-normal. (Proof: Consider the longest segment from C to C). Question: Let a "simple" double-normal be one that intersects C only at its endpoints. Must C have at least one simple double-normal? Prove or find a counterexample.
Consider an equilateral triangle T in the plane, and on each side build a 15-30-135 triangle, in the same sense, pointing into T. (Each side of T is the longest side of one 15-30-135 triangle.) Now remove the three original sides of T, leaving the 6 remaining sides of the obtuse triangles, which make a starlike hexagon H with 3-fold rotational symmetry. It's easy to check that if H is C^1-approximated by a smooth curve, then every double-normal segment (there are three) intersects the curve in its interior. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele