A Theorem of Lyapunov states that the range R(μ) of a non–atomic vector measure μ is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(μ) in terms of the Radon–Nikodym derivative of μ with respect to its total variation |μ|: namely R(μ) has an extremal face of dimension less or equal to k if and only if the set (x1, . . . , xk+1) such that ...

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of ...

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

In this paper, I provide the basis for a measureand integral-theoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno’s fuzzy integral – the seminormed fuzzy integral, or possibility integral –, a unified and consistent account can be given of many of the possibilistic results extant in the literature. The ...

In this note we described the robustness properties of optimal risksensitive controllers for quantum systems. We consider a quantum generalization of risk-sensitive criteria using the framework of (James, 2004). The robustness properties are derived by evaluating certain Radon-Nikodym derivatives of the quantum models and of the cost criteria. In addition to induced perturbations in the quantum...

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C (and C on every curve)...

In this paper we prove the differentiability of Lipschitz maps X → V , where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direct...

Abstract— A linear space-time block code (STBC) is a vector space spanned by its defining weight matrices over the real number field. We define a Quadratic Form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the ML decoding complexity of a STBC. It is shown that the ML decoding complexity is only a function of the weight matrices defining the code...

We study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. This is done under a relaxation of the classical immersion hypothesis, stated in terms of the changed probability measure, of which we characterize th...