A Six-Vertex Theorem for Bounding Normal Planar Curves

Abstract

We prove that the number of vertices of a smooth normal planar curve is at least 6, if it bounds a surface other than the disk.

Boundary torsion and convex caps of locally convex surfaces

“…Thus, if C intersects int H (D), then so does C i for large i, which is not possible. Hence ( 17) implies (16), which completes the proof of the first claim of the lemma.…”

“…Hence (17) C i ∩ int H (D) = ∅.Now note that int H (D) is open in ∂K.Thus, if C intersects int H (D), then so does C i for large i, which is not possible. Hence(17) implies(16), which completes the proof of the first claim of the lemma.To establish the second claim, suppose that C i → ∂K. Then int H (D) = ∅.…”

“…The above corollary also improves the theorem of Osserman [46] and dual results of Jackson [37] on the relation between the vertices and the inscribed/circumscribed circles of planar curves. Further, (7) yields the simply connected case of a theorem of Pinkall [50] who in 1987 had obtained the first version of the 4 vertex theorem for nonsimple curves which bound immersed surfaces in S 2 (or R 2 ), see also [16,67].…”

“…The rest of this paper is devoted to the proof of Theorem 1.1, although our methods yield a substantial refinement of it (Theorem 6. 16). The main inspiration here is a proof of the classical four vertex theorem for planar curves which dates back to H. Kneser [40] in 1922, and Bose [13] in 1932.…”

Analytic Non-Borel Sets and Vertices of Differentiable Curves in the Plane

“…Moreover, the subject of vertex theorems occupied the mind of several generations of mathematicians starting with the proof in 1909 of the original Four Vertex Theorem, concerning local extrema of the curvature, by the Indian mathematician S. Mukhopadhaya (see [20]). The MathSciNet contains several items concerning vertex theorems from which, according to their reviews, we selectively refer the reader to [1], [2], [4], [5], [7], [8], [9], [10], [13], [14], [15], [16], [23], [26], [28], [29], [32], and [33].…”