The width of embedded circles

Width is a classical geometric invariant of plane curves. It measures how narrow they are. Its definition, however, is based on Euclidean geometry and not easily generalisable beyond other constant curvature spaces. We will discuss how the variational theory of the Riemannian distance function, developed along the lines of Lusternik-Schnirelmann theory, can be used to define a meaningful notion of width for curves embedded in any complete Riemannian manifold, of any dimension. In particular, this definition allow to generalise another classical notion - curves of constant width - and to prove the existence of geodesics that meet two points of the curve in certain geometrically special configurations - for instance, in same cases, orthogonally. The talk will be based on joint work with Rafael Montezuma (UFC - Fortaleza) and Roney Santos (USP - São Paulo).