Characteristic functions freely generate measurable functions
Characteristic points, rectifiability and perimeter measure on stratified groups
We establish an explicit connection between the perimeter measure of an open set with boundary and the spherical Hausdorff measure restricted to , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of is less than or equal to up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo...
Characterization of local dimension functions of subsets of
For a subset and , the local Hausdorff dimension function of E at x is defined by where denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.
Characterization of optimal shapes and masses through Monge-Kantorovich equation
We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.
Characterization of σ-porosity via an infinite game
Let X be an arbitrary metric space and P be a porosity-like relation on X. We describe an infinite game which gives a characterization of σ-P-porous sets in X. This characterization can be applied to ordinary porosity above all but also to many other variants of porosity.
Characterizations of outer measures associated with lattice measures.
Characterizations of snowflake metric spaces.
Characterizations of sum form information measures on open domains.
Characterizing growth and form of fractal cities with allometric scaling exponents.
Checkerboards, Lipschitz functions and uniform rectifiability.
In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n > 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M < ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us...
Choquet integrals in potential theory.
This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to...
Choquet simplexes whose set of extreme points is -analytic
We construct a Choquet simplex whose set of extreme points is -analytic, but is not a -Borel set. The set has the surprising property of being a set in its Stone-Cech compactification. It is hence an example of a set that is not absolute.
Choquetova teorie kapacit
Christensen measurability of polynomial functions and convex functions of higher orders
Classes of measures which can be embedded in the simple symmetric random walk.
Classical hierarchies form a modern standpoint. Part III. BP-sets
Classical hierarchies from a modern standpoint. Part I. C-sets
Classical hierarchies from a modern standpoint. Part II. R-sets
Classification and extension by the transfinite induction
Classification of -ideals