Theorems about curves

Theorems about curves cover a vast range of topics in mathematics, particularly in geometry, calculus, and topology. Here are some key theorems and concepts associated with curves: 1. **Fermat's Last Theorem for Curves**: While Fermat's Last Theorem primarily concerns integers, there are generalizations and discussions about elliptic curves in number theory that relate deeply to the properties of curves.

Fenchel's theorem, often referred to in the context of convex analysis, deals with the correspondence between the convex functions and their subgradients. Specifically, it provides a characterization of convex functions through their conjugate functions.

The Fundamental Theorem of Curves is a concept in differential geometry that establishes a relationship between curves in Euclidean space and the properties of their curvature and torsion. While the specific formulation can vary depending on the context, generally, the theorem addresses the representation of a curve based on its intrinsic geometric properties.

The Jordan Curve Theorem is a fundamental result in topology, a branch of mathematics that studies properties of spaces that are preserved under continuous deformations. The theorem states that any simple closed curve in a plane (a curve that does not intersect itself and forms a complete loop) divides the plane into two distinct regions: an "inside" and an "outside.

Newton's theorem, often referred to as the "Newton's theorem on ovals," relates to the properties of an oval, particularly in the context of projective geometry and combinatorial geometry. The theorem essentially states that given a set of points in the plane, if these points are located on a smooth convex curve (an oval), then there exists a certain relationship concerning the tangents, secants, and other lines drawn from these points.

The Pestov–Ionin theorem is a result in the field of mathematical logic that deals with the preservation of certain properties in structures, particularly in the context of countable models. Although it is a specialized topic, the theorem itself is typically discussed within the framework of model theory, which studies the relationships between formal languages and their interpretations (or models).

The Tennis Ball Theorem is a concept from mathematics, specifically in the area of topology and geometry. It states that every point on the surface of a sphere can be connected to any other point on the sphere by a continuous path that lies entirely on the sphere's surface. This is often illustrated using the analogy of a tennis ball, which is a spherical object.