Around four vertices

The classical theorem of Mukhopadhyaya and Kneser [1], [2] asserts that a plane closed non-self-intersecting curve has no fewer than four vertices, that is, points of extremal curvature. Recently, this theorem has received renewed attention in connection with investigations in projective and symplectic topology [3]. In this note we consider a problem generalizing the four-vertex theorem, give two proofs of the theorem, and establish an assertion stating that a function on a circle not having small harmonics must have many zeros. 1. We consider a closed convex plane curve γ, and denote by L the set of points in the plane from which the two tangents to the curve have the same length. If γ is a curve in general position, L consists of non-intersecting embedded curves. The boundary points of L lie on γ and are vertices of it, and the ends of L, that is, the infinitely distant points, correspond to the oriented diameters of γ (chords perpendicular to γ). As in the vertex problem, the problem of equal tangents is projective-invariant: if x, y are points on γ that are points of contact of equal tangents to γ, and / is a Mobius transformation, then f(x) and/(y) are points of contact of equal tangents to the curve/(γ). This follows from the existence of a circle tangent to γ at χ and v. 2. Let A be a point outside γ, and let u and u be vectors along the right and left tangents from A to γ. We define two maps from the complement of the interior of γ to the plane: φ(.4)= ιι, ψ(Λ) = ν. A calculation establishes the following result. Proposition 1. (i) φ and ψ preserve area. (ii) Let B(r) be the circle of radius r and centre the origin. The sets φ~ ι (Β(ή) and Ψ~'(5(Γ)) have the same centre of gravity. To prove the four-vertex theorem, we need the following result. Lemma 2. The boundaries of two convex domains of the same area and with coincident centres of gravitv intersect in no fewer than four points. If there are two points in all, the moments of inertia of the domains about the lines joining the points are not the same. We denote by U(r) and V(r) the domains φ~ ι (Β(ή) and \|/~'(5(r)) complementing the interior of γ. The points of intersection of the boundaries of U(r) and V(r) lie in …