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 ...

Second-order optimization methods have the ability to accelerate convergence by modifying gradient through curvature matrix. There been many attempts use second-order for training deep neural networks. In this work, inspired diagonal approximations and factored such as Kronecker-factored Approximate Curvature (KFAC), we propose a new approximation Fisher information matrix (FIM) called Trace-re...

In this work, we propose to apply trust region optimization to deep reinforcement learning using a recently proposed Kronecker-factored approximation to the curvature. We extend the framework of natural policy gradient and propose to optimize both the actor and the critic using Kronecker-factored approximate curvature (K-FAC) with trust region; hence we call our method Actor Critic using Kronec...

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in cases of both symmetric and non-symmetric connection. Although curvature-flatness idea classic for connection, our main theorems about flatness solutions completely new, leaving a while point view geometry entering PDEs. second int...

We construct, for any " good " Cantor set F of S n−1 , an immersion of the sphere S n with set of points of zero Gauss-Kronecker curvature equal to F ×D 1 , where D 1 is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do Carmo-Elbert and Barbosa-Fukuoka-Mercuri.

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...

In this paper we study the Dirichlet problem for some Monge-Ampère type equations on S, which naturally arise in some geometric problems. The result then is applied to prove the existence of hypersurfaces in R of prescribed Gauss-Kronecker curvature and with fixed boundary.