In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds
$Mto B$, $B$ is connected. For $varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_varepsilon$. The main result says that $lim_{varepsilonto +0} V_i(M_varepsilon)=chi(Z) V_i(B)$, where $chi(Z)$ is the Euler characteristic of a fiber of the submersion. This result is consistent with more general open conjectures on convergence of intrinsic volumes formulated previously by the author.
Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbatys theorem to obtain a
few apparently new inequalities for mixed volumes of convex bodies.
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functio
ns which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the authors irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.
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 v
olumes 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.
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of
continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized v
aluations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.
We prove a $C^0$ a priori estimate on a solution of the quaternionic Calabi problem on an arbitrary compact connected HKT-manifold. This generalizes earlier works where this result was proven under certain extra assumptions on the manifold.
The goal of this paper is to describe the $alpha$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $alph
a$ the $alpha$-cosine transform is a composition of the $(alpha+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $alpha$ except one we interpret the $alpha$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $alpha$, which is $-(min{i,n-i}+1)$, is still an open problem.
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former
space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
