Four-vertex theorem

Every smooth, simple, closed convex plane curve has at least four vertices, points where curvature reaches a local maximum or minimum.

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Curvature: A measure of how rapidly a curve’s direction changes along its length. Vertices are defined as local extrema of this quantity along the curve.
Plane curve: A continuous path in the plane, often specified by smooth coordinate functions. The theorem concerns closed curves lying in the Euclidean plane.
Sylvester–Cayley theorem: A nineteenth-century result asserting that a smooth closed plane curve has at least four vertices. Its formulation is an early source of the four-vertex result.
Circle: The set of plane points at a fixed distance from a center. Every point has the same curvature, so every point is both a local maximum and minimum under non-strict conventions.
Constant width: A property of a convex plane body whose distance between parallel supporting lines is the same in every direction. Curvature extrema help analyze smooth curves of constant width.
Support function: A function recording the signed distance from a chosen origin to a supporting line of a convex body. It represents a convex curve in a way that makes its curvature accessible through derivatives.
Convex set: A set containing the whole line segment between any two of its points. Convexity prevents inward indentations that would undermine the theorem’s conclusion.
Otto Toeplitz: A German mathematician known for work in analysis and geometry, including the four-vertex problem. Toeplitz gave a celebrated geometric proof for convex curves.
Ellipse: A closed plane curve whose points have a constant sum of distances from two foci. A noncircular ellipse has exactly four curvature extrema.
Blaschke–Lebesgue theorem: The theorem that the Reuleaux triangle has least area among plane bodies of a given constant width. It illustrates geometric consequences of studying convex curves beyond vertex counting.