Blaschke–Lebesgue theorem

The Blaschke–Lebesgue theorem states that among planar convex bodies of a fixed constant width, the Reuleaux triangle has the least area.

Constant-width curve: A plane curve whose width is the same in every direction. The theorem compares convex bodies within this geometric class.
Reuleaux triangle: A constant-width shape formed by three circular arcs centered at the vertices of an equilateral triangle. It is the shape attaining the least area in the theorem.
Disk: The planar region consisting of all points within a fixed distance of a center. It has the same width as the Reuleaux triangle but greater area.
Wilhelm Blaschke: An Austrian mathematician known for work in geometry and topology. His name is attached to the theorem on least area at fixed width.
Convex body: A compact convex subset of Euclidean space with nonempty interior. The theorem concerns planar convex bodies rather than arbitrary shapes.
Barbier's theorem: The perimeter of every planar convex body of constant width w is πw. It shows that the theorem’s competing bodies already share the same perimeter.
Reuleaux polygon: A constant-width shape whose boundary is composed of circular arcs centered at vertices of a polygon. The Reuleaux triangle is the simplest and smallest-area member of this family.
Henri Lebesgue: A French mathematician whose work shaped measure theory and integration. His name is paired with Blaschke’s in the theorem’s standard title.
Width (geometry): The distance between two parallel supporting lines of a convex set. Equal width in every direction is the constraint in the theorem.
Minkowski addition: An operation adding two sets by taking all sums of one point from each set. Support-function methods connect constant-width constraints to geometric transformations.

Linked from 1 page

Four-vertex theoremRelated: It illustrates geometric consequences of studying convex curves beyond vertex counting.