  For any closed smooth Riemannian manifold H. Weyl has defined a sequence of
numbers called today intrinsic volumes. They include volume, Euler
characteristic, and integral of the scalar curvature. We conjecture that
absolute values of all intrinsic volumes are bounded by a constant depending
only on the dimension of the manifold, upper bound on its diameter, and lower
bound on the sectional curvature. Furthermore we conjecture that intrinsic
volumes can be defined for some (so called smoothable) Alexandrov spaces with
curvature bounded below and state few of the expected properties of them,
particularly the behavior under the Gromov-Hausdorff limits. We suggest
conjectural compactifications of the space of smooth closed connected
Riemannian manifolds with given upper bounds on dimension and diameter and a
lower bound on sectional curvature to which the intrinsic volumes extend by
continuity. We discuss also known cases of some of these conjectures.
