A Sufficient Condition for a Curve to Belong to Boundary of its Own Convex Hullin An

Abstract

The paper discusses some generalizations of the convex curve and the relationship between the classes of curves they define. The main result concerns the curves in n-dimensional affine space An: a non-degenerate curve in An lies on the boundary of its convex hull if any hyperplane intersects it in at most n points. It is justified by the theory of convex sets. This statement generalizes an earlier (2014) result of the author concerning the curves that are compact sets in An. V. Gustin proved in 1947 a stronger theorem for a Euclidean space En of dimension n: a connected set intersected by any hyperplane at no more than n points is a simple continuous curve lying on the boundary of a convex set. Note that in a projective space of dimension n the requirement that the curve intersects all planes at no more than n points is included in the definition of a convex curve. Thus, the established fact, together with the result of V. Gustin, shows the contiguity of the concepts of convexity in all three spaces.