points in three-dimensional space, a neighborhood of a spanning tree of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the obstacle problem of finding a minimum-energy surface above a given shape can have the convex hull as their solution. For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of the objects. The definition using intersections of convex sets may be extended to non-Euclidean geometry, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids. Equivalence of definitions It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing , for every ? However, the second definition, the intersection of all convex sets containing , is well-defined. It is a subset of every other convex set that contains , because is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing . Therefore, the first two definitions are equivalent. Each convex set containing must (by the assumption that it is convex) contain all convex combinations of points in , so the set of all convex combinations is contained in the intersection of all convex sets containing . Conversely, the set of all convex combinations is itself a convex set containing , so it also contains the intersection of all convex sets containing , and therefore the second and third definitions are equivalent. In fact, according to Carathéodory's theorem, if is a subset of a -dimensional Euclidean space, every convex combination of finitely many points from is also a convex combination of at most points in . The set of convex combinations of a -tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of belongs to a simplex whose vertices belong to , and the third and fourth definitions are equivalent. Upper and lower hulls In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex