Related papers: Variations on the capacitary inradius

The notion of the magnitude of a metric space was introduced by Leinster in [8] and developed in [10], [9], [11] and [16], but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we…

We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy…

We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out the similarities and distinctions…

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between…

The intrinsic unsharpness of a quantum observable is studied by introducing the notion of resolution width. This quantification of accuracy is shown to be closely connected with the possibility of making approximately repeatable…

Contains a further development of the V.G. Maz'ya concept of polynomial capacities.A different different approach is given together with new results. The primary application is for Hardy inequalities. This is treated in a later report. Here…

In this paper, we examine some shape functionals, introduced by P\'olya and Makai, involving the torsional rigidity and the first Dirichlet-Laplacian eigenvalue for bounded, open and convex sets of $\mathbb{R}^n$. We establish new…

On a convex set, we prove that the Poincar\'e-Sobolev constant for functions vanishing at the boundary can be bounded from above by the ratio between the perimeter and a suitable power of the $N-$dimensional measure. This generalizes an old…

We solve the commutant lifting and interpolation problems in the setting of the Hardy space and Schur functions on the open unit ball of $\mathbb{C}^n$. Our solutions also signify the role of inner functions on the unit ball, objects whose…

We study mappings that satisfy the inverse modulus inequality of Poletsky type in a fixed domain. It is shown that, under some additional restrictions, the image of a ball under such mappings contains a fixed ball uniformly over the class.…

In this article we present both the discrete and the integral form of Cauchy-Bunyakovsky-Schwarz (CBS) inequality, some important generalizations in the n-dimensional Euclidean space and in linear subspaces of it, as well as the…

We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…

We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…

We consider an overdetermined problem arising in potential theory for the capacitary potential and we prove a radial symmetry result.

We characterize the fractional Sobolev inequality with fractional isocapacitary and isoperimetric inequalities. We give a sufficient condition and examples so that the fractional capacity of the closure of an open set is bounded above by…

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…

We present several applications of matrix-theoretic inequalities to the magnitude of metric spaces. We first resolve an open problem by showing that the magnitude of any finite metric space of negative type is less than or equal to its…

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that…

We study geometric characterizations of the Poincar\'{e} inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of…