Length of convex curve

The following is a result of Bean after discussing with him some time ago.

Theorem 1 (Bean’s theorem, 2010, Polygonal version)
Suppose {\gamma} is a convex polygon contained in the interior of the unit circle {\{(x, y)\in {\mathbb R}^2: x^2+y^2=1\}}, then the length of {\gamma} is not more than the length of the unit circle, i.e. {2\pi }.

Lemma 2 Any two outward going normals of a convex polygon {\gamma} (or a convex closed curve) will not meet.

Proof: Suppose not, let {p} be the intersection point of the normals starting from {q} and {r} respectively. Since any triangle cannot have more than two right angles, we can assume that {\angle pqr<\pi/2}. But then, since {\gamma} is convex with {\vec qp} being outward pointing, {\gamma} must lies on the half plane {H=\{x: (x-q)\cdot (p-q)\leq 0\}}, but clearly {r\in\gamma} which does not belong to {H} (consider the line perpendicular to {pr} at that point! {q} lies “under” it!), this is a contradiction. \Box

Proof of Bean’s theorem, polygonal version:
Fix a side {pq} of the polygon, push it in the outward normal direction until (at least) one endpoint of it, say {p'} of the translated segment meet the unit circle, let {p', q'} be the endpoints of this translated segment. There are two cases,
case (i): Suppose both {p', q'} are on the unit circle, then clearly the length of the arc {\alpha(p, q)} connecting {p', q'} (corresponding to the side {pq}) must have length greater than {pq}.
Case (ii): Suppose only {p'} is on the unit circle. Extend {qq'} in the outward normal direction to meet the unit circle at {r'}, then by Pythagoras theorem, the length of {pq} must be less than that of {p'r'} which in turn is less than the length of the arc joining {p' } and {r'} . Let us also call this arc {\alpha (p,q )}.
We are done if we can show that {\alpha (p, q)} and {\alpha(s, t)} are disjoint (at least in its interior) if {pq} and {st} are different side of the polygon. But this is exactly the statement of the above lemma. So we have completed the proof. \Box

Remark 1

  1. 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 {2\pi}.
  2. For another proof, see here.

  3. 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(!?).

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One Response to Length of convex curve

  1. Hon Leung says:

    For Bean’s theorem, should it be “less than” instead of “not less than”?
    [Corrected, thanks! -KKK]
    Besides, it seems similar proof works for platonic solids inside unit sphere, replacing the length to be surface area. [Yes! But I always want to work in the simplest nontrivial case, which the idea is the clearest. All the possible generalizations are left to the readers. -KKK]

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