Sufficient Conditions for Convexity and Affineness of a Continuous Map

The article establishes a criterion for the convexity of a closed set in a topological vector space: a closed set in a topological vector space is convex if and only if every segment with endpoints in this set contains at least one more point of this set. It generalizes a similar result established earlier for reflexive Banach spaces. It is used to prove a sufficient condition for the planarity of a k-dimensional manifold in an n-dimensional affine space An: if each chord of a k-dimensional surface, which is a closed set, contains some other point of the surface except its endpoints, then the surface is a k-dimensional plane or its convex subset with non-empty interior relative to this plane. These 2 statements together with the closed graph theorem are used to establish sufficient convexity and affine conditions for a continuous multivariable function. It also allows the Jensen functional equation of multivariable functions in the class of continuous functions to be solved in a new way. Proof methods are topological.