The following is a result of Bean after discussing with him some time ago.
Theorem 1 (Bean’s theorem, 2010, Polygonal version)
Suppose is a convex polygon contained in the interior of the unit circle , then the length of is not more than the length of the unit circle, i.e. .
Lemma 2 Any two outward going normals of a convex polygon (or a convex closed curve) will not meet.
Proof: Suppose not, let be the intersection point of the normals starting from and respectively. Since any triangle cannot have more than two right angles, we can assume that . But then, since is convex with being outward pointing, must lies on the half plane , but clearly which does not belong to (consider the line perpendicular to at that point! lies “under” it!), this is a contradiction.
Proof of Bean’s theorem, polygonal version:
Fix a side of the polygon, push it in the outward normal direction until (at least) one endpoint of it, say of the translated segment meet the unit circle, let be the endpoints of this translated segment. There are two cases,
case (i): Suppose both are on the unit circle, then clearly the length of the arc connecting (corresponding to the side ) must have length greater than .
Case (ii): Suppose only is on the unit circle. Extend in the outward normal direction to meet the unit circle at , then by Pythagoras theorem, the length of must be less than that of which in turn is less than the length of the arc joining and . Let us also call this arc .
We are done if we can show that and are disjoint (at least in its interior) if and are different side of the polygon. But this is exactly the statement of the above lemma. So we have completed the proof.
- Actually the assertion is also true for any smooth convex closed curve (Bean’s theorem, 2010, smooth version), or even piecewise smooth convex closed curve inside the unit circle. Actually, if the above result is true, since it is well-known that any closed convex closed curve is rectifiable, i.e. it length can be approximated by finite line segment, and in particular by convex polygon since it is convex, it then can be proved that the length of any piecewise smooth convex closed curve inside the unit circle has length not more than .
- For another proof, see here.
- I think Paul Lai have another proof, using Fourier Analysis. (Ask him!). Also, the equality holds only when it is the unit circle,
but Stein seems to disagree(!?).