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By Luther Pfahler Eisenhart

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Proof of corollary. 6). 1 Definition. A regular plane curve c: I ~ 1R2 is convex if, for all to EI, the curve lies entirely on one side of the tangent at c(to). 2 Theorem (A characterization of convex curves). Let c: I ~ 1R2 be a simple closed regular plane curve. Then c is convex if and only if one of the following conditions are true: Ie(t) ~ 0, alltEI or Ie(t) :s; 0, ali tEl. Remarks. i) If one of the above conditions hold then an orientation-reversing change of variables will produce the other.

Osserman, R. Isoperimetric and related inequalities. Proc. AMS Symp. in Pure and Applied Math. XXVII, Part 1, 207-215. ,. Dubins, L. E. On curves of minimal length with constraint an average curvature and prescribed initial and terminal positions and tangents. Amer. J. , 79, 497-516(1957). 11 Fenchel, W. Ober KrUmmung und Wendung geschlossener Raumkurven. Math. Ann. 101, 238-252 (1929). Ce. also Fenchel, W. On the differential geometry of c10sed space curves. Bull. Amer. Math. , 57, 44-54 (1951), ar Chem [A5].

I«to) = O, to Ei. If I<(t) = const, tI :<;; t :<;; t 2, alI these tare vertices. 4 Theorem (Four vertex theorem). A convex, simple, c/osed smooth plane curve has at least four vertices. Remark. The theorem is true without the convexity hypothesis (although it is harder to prove). 4 Exercises and Some Further Results (due to G. Herglotz)2 Step 1. Since K(t) has a maximum and a minimum on 1, c(t) has at least two vertices. Without loss of generality, we may as sume that c is parameterized by arc length and that K(t) has a minimum at t = Oand a maximum at to, O < to < w, where I = [O, w].

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An introduction to differential geometry, with use of the tensor calculus by Luther Pfahler Eisenhart

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