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Julià Cufí

Publications and source records attributed to Julià Cufí.

15 recordsLinked to original sources
arXiv2026

On the Minkowski deficit for surfaces of revolution

By studying the classical Minkowski's inequality, $4πA\leq M^{2}$, relating area and mean curvature, in the particular case of convex surfaces of revolution, one is led to a functional inequality that holds for general functions. We state and prove this inequality obtaining a new proof of Minkowski's inequality in this particular case and we establish a sharp inequality between the Minkowski deficit of the surface and the isoperimetric deficit of any meridian curve.

math.DG↗
arXiv2024

On some relations between the Perimeter, the Area and the Visual Angle of a Convex Set

We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton's formula is the unique universal formula relating the visual angle, length and area. After that we give a characterization of convex sets of constant width by means of the behaviour of its isotopic sets at infinity. Also for this class of convex sets we prove that the existence of an isotopic circle is enough to ensure that the considered set is a disc.

math.DG↗
arXiv2023

A historical review of the Cauchy-Riemann equations and the Cauchy Theorem

In this article we take a historical tour through the Cauchy-Riemann equations and their relationship with Cauchy's theorem on the independence with respect to the path of the integral of a holomorphic function. Because of its importance we do a detailed and updated study of the contributions of d'Alembert and Euler to these topics. We also review the Cauchy works about the passage from the real to imaginary by paying attention to some arguments he uses that are not clear enough. At the end we comment briefly the evolution of Green's formula and its relation with the above problems.

math.HO↗
arXiv2022

The precise representative for the gradient of the Riesz potential of a finite measure

Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

math.CA↗
arXiv2021

Integral Geometry of pairs of planes

We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke.

math.DG↗
arXiv2019

Integral Geometry about the visual angle of a convex set

In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.

math.DG↗
arXiv2019

A discrete approach to Wirtinger's inequality

Considering Wirtinger's inequality for piece-wise equipartite functions we find a discrete version of this classical inequality. The main tool we use is the theorem of classification of isometries. Our approach provides a new elementary proof of Wirtinger's inequality that also allows to study the case of equality. Moreover it leads in a natural way to the Fourier series development of $2π$-periodic functions.

math.CA↗
arXiv2017

On square functions with independent increments and Sobolev spaces on the line

We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy-Sobolev spaces $\dot H^1_α$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.

math.CA↗
arXiv2017

A note on Hurwitz's inequality

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly positive lower bounds for the deficit $π|F_{e}|-Δ$, where $Δ=L^{2}-4πF$. These bounds involve wether the visual angle of $Γ$ or the pedal curve associated to $K$ with respect to the Steiner point of $K$ or the $\mathcal{L}^{2}$ distance between $K$ and the Steiner disk of $K$. For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of $3, 4$ or $5$ cusps or the Minkowski sum of this kind of sets.

math.DG↗
arXiv2017

Differentiability properties of Riesz potentials of finite measures and non-doubling Calderón-Zygmund theory

We study differentiability properties of Riesz potentials of finite Borel measures in dimension d larger than 2. The Riesz kernel has homogeneity 2-d. In dimension 2 we consider logarithmic potentials. We introduce a notion of differentiability in the capacity sense, capacity being Newtonian capacity in dimension larger than 2 and Wiener capacity in the plane. It turns out that differentiability in the capacity sense at a point is related to the existence of principal values of the measure with respect to the vector valued Riesz potential x/|x|^d of homogeneity 1-d. This leads to Calderón-Zygmund theory for non-doubling measures. We prove that the Riesz potential of a finite Borel measure is differentiable in the capacity sense except for a set of zero C^1-harmonic capacity. This result is sharp. Surprisingly in the plane there are two distinct notions of differentiability in the capacity sense. For each of them we obtain the best possible result on the size of the exceptional set in terms of Hausdorff measures. We obtain for dimension larger than 2 results on Peano second order differentiability in the capacity sense with exceptional sets of zero Lebesgue measure. Finally, we apply our results to find a new proof of the fact that equilibrium measure is singular with respect to Lebesgue measure.

math.CA↗
arXiv2016

A lower bound for the isoperimetric deficit

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.

math.DG↗
arXiv2011

Characterizing abelian admissible groups

By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block diagonalization result for commuting real valued matrices. We then reduce the question of deciding admissibility to the subclass of connected and simply connected groups, and derive a general admissibility criterion for exponential solvable matrix groups. For abelian matrix groups with real spectra, this yields an easily checked necessary and sufficient characterization of admissibility. As an application, we sketch a procedure how to check admissibility of a matrix group generated by finitely many commuting matrices with positive spectra.

math.FA↗