Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be globally definable one parameter family of $C^2$-hypersurfaces $\mathbb{R}^n$. Upon defining notion generalized critical value for such we show that functions $s \to |K(s)|$ and $s\to K(s)$, respectively total absolute Gauss-Kronecker curvature $T_s$, are continuous in any neighb...

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...

Let f(x) be a given positive function in Rn+1. In this paper we consider the existence of convex, closed hypersurfaces X so that its GaussKronecker curvature at x ∈ X is equal to f(x). This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the ...

For convex hypersurfaces in the affine space $\mathbb{A}^{n+1}$ ($n\geq2$), A.-M.\ Li introduced notion of $\alpha$-normal field as a generalization normal field. By studying Monge-Amp\`ere equation with gradient blowup boundary condition, we show that regular domains $\mathbb{A}^{n+1}$, defined respect to proper cone and satisfying some regularity assumption if $n\geq3$, are foliated by comple...

The L-cosine transform of an even, continuous function f ∈ Ce(S ) is defined by: H(x) = ∫ S |〈x, ξ〉|f(ξ) dξ, x ∈ R. It is shown that if p is not an even integer then all partial derivatives of even order of H(x) up to order p + 1 (including p + 1 if p is an odd integer) exist and are continuous everywhere in R\{0}. As a result of the corresponding differentiation formula, we show that if f is a...

LetK be a d-dimensional convex bodywith a twice continuously differentiable boundary and everywhere positive Gauss–Kronecker curvature. Denote byKn the convex hull of n points chosen randomly and independently fromK according to the uniform distribution. Matching lower andupper bounds are obtained for the orders ofmagnitudeof the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1, ....