In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter \(\Omega \) expressed the with respect to measure \(P(\Omega ,\cdot )\) which equivalent one-codimensional Hausdorff restricted reduced boundary \). The same result has been proved in an abstract Wiener typically infinite-dimensional where surface considered spherical Hausdorff–Gauss \(\math...

A model of coherent upper conditional prevision for bounded random variables is proposed in a metric space. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff...

If S is an infinite sequence over a finite alphabet Σ and β is a probability measure on Σ, then the dimension of S with respect to β , written dimβ (S), is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension dim(S) when β is the uniform probability measure. This paper shows that dimβ (S) and its dual Dimβ (S), the strong dimension of S with...

Let Σ : M → 2 \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, ‖·‖). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y ). In an abstract setting this problem is posed as the extremal problem supt∈M ...

For a map S : X → X and an open connected set (= a hole) H ⊂ X we define JH(S) to be the set of points in X whose S-orbit avoids H. We say that a hole H0 is supercritical if (i) for any hole H such that H0 ⊂ H the set JH(S) is either empty or contains only fixed points of S; (ii) for any hole H such that H ⊂ H0 the Hausdorff dimension of JH(S) is positive. The purpose of this note is to complet...

‎it is well known that every (real or complex) normed linear space $l$ is isometrically embeddable into $c(x)$ for some compact hausdorff space $x$‎. ‎here $x$ is the closed unit ball of $l^*$ (the set of all continuous scalar-valued linear mappings on $l$) endowed with the weak$^*$ topology‎, ‎which is compact by the banach--alaoglu theorem‎. ‎we prove that the compact hausdorff space $x$ can ...

We present a study of the Hausdorff Core problem on simple polygons. A polygon Q is a k-bounded Hausdorff Core of a polygon P if P contains Q, Q is convex, and the Hausdorff distance between P and Q is at most k. A Hausdorff Core of P is a k-bounded Hausdorff Core of P with the minimum possible value of k, which we denote kmin. Given any k and any ε > 0, we describe an algorithm for computing a...

We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class gauge functions. This yields lower and upper bounds in terms general notions capacity Hausdorff measure, respectively, therefore extending the classical estimates Bessel-Riesz $\gamma$-dimensional measure. apply to system linear stochastic partial differen...

We show the existence result of viable solutions to the differential inclusion ẋ(t) ∈ G(x(t)) + F (t, x(t)) x(t) ∈ S on [0, T ], where F : [0, T ] × H → H (T > 0) is a continuous set-valued mapping, G : H → H is a Hausdorff upper semi-continuous set-valued mapping such that G(x) ⊂ ∂g(x), where g : H → R is a regular and locally Lipschitz function and S is a ball, compact subset in a separable H...

It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) > (1− ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.