On Young Hulls of Convex Curves in R 2n

For a convex curve in an even-dimensional aane space we introduce a series of convex domains (called Young hulls) describe their structure and give a formulas for the volume of the biggest of these domains. A simple smooth curve in R m is called convex if the total multiplicity of its intersection with any aane hyperplane does not exceed m. Note that each convex curve is nondegen-erate, i.e. has a nondegenerate osculating m-frame at each point. Nondegenerate curves in projective spaces have naturally arisen in many classical geometrical problems as well as in problems related to linear ordinary diierential equations, see e.g. 4,7,13]. The global topological properties of the spaces of such curves were used in the enumeration of symplectic leaves of the Gelfand-Dickey bracket, 5]. The set of all nondegenerate curves (say, with xed osculating ags at both endpoints) contains the subset of convex curves which minimize the maximal possible number of intersection points with hyperplanes. Convex curves correspond to a special class of linear ODE called disconjugate. Their properties were studied, for example, in the classical papers 3,6,9] in connection with Sturmian theory and functional analysis. Recently V. Arnold proposed a generalization of Sturmian theory which in the simplest treated case deals with the estimation of the number of attening points of a curve in R 2n+1 which projects on a