In this paper we give some geometric characterizations of locally convex hypersurfaces. In particular, we prove that for a given locally convex hypersurface M with boundary, there exists r > 0 depending only on the diameter of M and the principal curvatures of M on its boundary, such that the r-neighbourhood of any given point on M is convex. As an application we prove an existence theorem for ...

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a …nite union of convex sets (i.e. the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The conv...

The paper studies convex Lyapunov functions for differential and difference inclusions with right-hand sides given by convex processes, that is, by set-valued mappings the graphs of which are convex cones. Convex conjugacy between weak Lyapunov functions for such inclusions and Lyapunov functions for adjoint inclusions is established. Asymptotic stability concepts are compared and the existence...

In this paper, a new definition of locally convex L-topological vector spaces is given. The relationship between this new definition and the previous definition of locally convex L-topological vector spaces given by Yan and Fang in 1999 is investigated. Moreover, the concept of generalized L-fuzzy semi-norm is introduced. By using a family of generalized L-fuzzy semi-norms, a characterization o...

We establish the following max-plus analogue of Minkowski’s theorem. Any point of a compact max-plus convex subset of (R∪{−∞})n can be written as the max-plus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set ca...

By using an axiomatic approach we formalize the concept of size index for closed convex cones in the Euclidean space R. We review a dozen of size indices disseminated through the literature, commenting on the advantages and disadvantages of each choice. Mathematics Subject Classification: 28A75, 51M25, 52A20, 52A40.

On the interior of a regular convex cone K ⊂ R there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K with parameter of order O(n), the universal barrier. This barrier is invariant with respect to the unimodular automorphism su...

Estimation of a multivariate normal mean is considered when the latter is known to belong to a convex polyhedron. It is shown that shrinking the maximum likelihood estimator towards an appropriate target can reduce mean squared error. The proof combines an unbiased estimator of a risk difference with some geometrical considerations. When applied to the monotone regression problem, the main resu...