Carathéodory conjecture

The conjecture states that every closed, smooth, strictly convex surface in three-dimensional space has at least two umbilic points, where its principal curvatures are equal.

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Umbilic point: A point on a smooth surface where the two principal curvatures are equal. These are the points whose existence the conjecture requires.
Convex surface: A surface bounding a convex body, with every segment between body points contained in the body. Convexity is a central restriction on the surfaces covered by the conjecture.
Hans Hamburger: A German mathematician who published an attempted proof of the Carathéodory conjecture in 1924. His work is an early landmark in the conjecture's proof history.
Ellipsoid: A quadric surface obtained by scaling a sphere along three perpendicular axes. A triaxial ellipsoid has four umbilics, illustrating that the minimum need not be exact.
Line field: An assignment of an unoriented one-dimensional subspace of a tangent space at each point, except possibly singularities. Principal directions form line fields rather than globally oriented vector fields.
Principal curvature: An extremal normal curvature of a surface at a point, measured in a principal direction. Equality of the two principal curvatures defines an umbilic.
Gaussian curvature: The product of a surface's two principal curvatures at a point. Strict convexity ensures positive Gaussian curvature under the usual smooth-surface convention.
Constantin Carathéodory: A Greek-German mathematician known for contributions to analysis, calculus of variations, and thermodynamics. He posed the problem about umbilics on convex surfaces.
Sphere: The set of points at a fixed distance from a center in Euclidean space. Every point on a sphere is umbilic, so it satisfies the claim non-isolatedly.
Index of a line field singularity: A half-integer or integer measuring the rotation of a line field around an isolated singularity. The possible indices constrain how many umbilics a convex sphere can have.