1. The following question, raised originally by S. Mazur, appears in S. M. Ulam's collection of mathematical problems [1 ]: does there exist a closed convex surface whose plane sections give all plane closed convex curves, up to affinities? While this problem is apparently still unsolved the answer is almost certainly negative. At least three different extensions of the problem could be considered: (1) to allow the plane sections an equivalence up to a larger class, or perhaps a larger group, of transformations than the affinities, (2) to ask that the set of all plane sections of the surface should only contain a sufficiently large subset of the set of all closed convex curves, for instance, the set of all convex polygons of given diameter or perimeter, all analytic ovals of fixed length, or all ovals of given constant width, and (3) to generalize the concept of a plane section. This note is concerned with the last possibility. Instead of plane sections of a surface one considers its limit sections. Roughly speaking, these are limits of sequences of magnified sections of a surface by sequences of planes converging to a supporting plane. For instance, if a strictly convex surface is sufficiently regular the limit section will always be an ellipse with axes proportional to the square roots of the principal radii of curvature. Thus the limit section is a generalization of Dupin's indicatrix. The following notation will be used: C and D will denote curves, other capital letters will usually denote surfaces, P will be reserved for planes, small letters will stand for points, and small Greek letters will be non-negative constants. A surface (curve) will always mean a closed strictly convex surface (a closed plane convex curve). A part of a surface cut off by a plane will be called a cap. The set of all curves will be denoted by cl.