Papers on Convex Geometry
2018
J. Cufí, E. Gallego, A. Reventós
A note on Hurwitz’s inequality Journal Article
In: Journal of Mathematical Analysis and Applications, vol. 458, pp. 436-451, 2018.
@article{1e,
title = {A note on Hurwitz’s inequality },
author = {J. Cufí, E. Gallego, A. Reventós},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/JMAA.pdf},
year = {2018},
date = {2018-10-11},
journal = {Journal of Mathematical Analysis and Applications},
volume = {458},
pages = {436-451},
abstract = {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
L2 − 4πF ≤ π|Fe|, where Fe 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|Fe|−Δ, where Δ = L2 −4πF. These bounds involve either the visual angle
of Γ or the pedal curve associated to K with respect to the Steiner point of K or
the L2 distance between K and the Steiner disk of K. For compact convex sets of
constant width Hurwitz’s inequality can be improved to L2 −4πF ≤ 4 π|Fe|. In this 9
case we also get strictly positive lower bounds for the deficit 4 π|Fe| − Δ. For each 9
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.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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
L2 − 4πF ≤ π|Fe|, where Fe 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|Fe|−Δ, where Δ = L2 −4πF. These bounds involve either the visual angle
of Γ or the pedal curve associated to K with respect to the Steiner point of K or
the L2 distance between K and the Steiner disk of K. For compact convex sets of
constant width Hurwitz’s inequality can be improved to L2 −4πF ≤ 4 π|Fe|. In this 9
case we also get strictly positive lower bounds for the deficit 4 π|Fe| − Δ. For each 9
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.
K of area F, Hurwitz found an upper bound for the isoperimetric deficit, namely
L2 − 4πF ≤ π|Fe|, where Fe 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|Fe|−Δ, where Δ = L2 −4πF. These bounds involve either the visual angle
of Γ or the pedal curve associated to K with respect to the Steiner point of K or
the L2 distance between K and the Steiner disk of K. For compact convex sets of
constant width Hurwitz’s inequality can be improved to L2 −4πF ≤ 4 π|Fe|. In this 9
case we also get strictly positive lower bounds for the deficit 4 π|Fe| − Δ. For each 9
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.
2016
J. Cufí, A. Reventós
A lower bound for the isoperimetric deficit Journal Article
In: Elemente der Mathematik, vol. 71, pp. 156-167, 2016.
@article{1f,
title = {A lower bound for the isoperimetric deficit},
author = {J. Cufí, A. Reventós },
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/Cufi-Reventos.pdf},
year = {2016},
date = {2016-10-01},
journal = {Elemente der Mathematik},
volume = {71},
pages = {156-167},
abstract = {Fu ̈r jede Figur K in der Ebene mit Umfang L und Fla ̈che F gilt die isoperimetrische Ungleichung := L 2 − 4π F ≥ 0. Gleichheit gilt genau fu ̈ r Kreise. Hurwitz gelang 1902 nicht nur ein eleganter Beweis der isoperimetrischen Ungleichung mit Hilfe von Fourier-Reihen, er bewies zudem eine obere Schranke fu ̈r das isoperimetrische Defizit , indem er die Evolute der Kurve ins Spiel brachte. 1920 fand Bonnesen eine un- tere Schranke fu ̈r , na ̈mtlich π(R − r)2 ≤ , wobei R und r den Um- respektive den Inkreisradius der Randkurve C der betrachteten Figur K bezeichnen. In der vor- liegenden Arbeit wird eine andere untere Schranke fu ̈r bewiesen: Diese ergibt sich aus der Differenz der von C umrandeten Fla ̈che und der Fla ̈che welche die Pedalkurve von C bezu ̈glich des Steiner-Punktes von C einschliesst. Das Resultat verbessert damit Abscha ̈tzungen z.B. von Groemer. Es wird zudem bestimmt, fu ̈r welche Kurven die neue Abscha ̈tzung scharf ist.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Fu ̈r jede Figur K in der Ebene mit Umfang L und Fla ̈che F gilt die isoperimetrische Ungleichung := L 2 − 4π F ≥ 0. Gleichheit gilt genau fu ̈ r Kreise. Hurwitz gelang 1902 nicht nur ein eleganter Beweis der isoperimetrischen Ungleichung mit Hilfe von Fourier-Reihen, er bewies zudem eine obere Schranke fu ̈r das isoperimetrische Defizit , indem er die Evolute der Kurve ins Spiel brachte. 1920 fand Bonnesen eine un- tere Schranke fu ̈r , na ̈mtlich π(R − r)2 ≤ , wobei R und r den Um- respektive den Inkreisradius der Randkurve C der betrachteten Figur K bezeichnen. In der vor- liegenden Arbeit wird eine andere untere Schranke fu ̈r bewiesen: Diese ergibt sich aus der Differenz der von C umrandeten Fla ̈che und der Fla ̈che welche die Pedalkurve von C bezu ̈glich des Steiner-Punktes von C einschliesst. Das Resultat verbessert damit Abscha ̈tzungen z.B. von Groemer. Es wird zudem bestimmt, fu ̈r welche Kurven die neue Abscha ̈tzung scharf ist.
2015
J. Cufí, A. Reventós; C. Rodríguez
Curvature for Polygons Journal Article
In: The American Mathematical Monthly, vol. 122, no. 4, pp. 332-337, 2015.
@article{1h,
title = {Curvature for Polygons},
author = {J. Cufí, A. Reventós and C. Rodríguez},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/amer.math_.monthly.122.04.332.pdf},
year = {2015},
date = {2015-10-01},
journal = {The American Mathematical Monthly},
volume = {122},
number = {4},
pages = {332-337},
abstract = {Using a notion of curvature at the vertices of a polygon, we prove an inequality involving the length of the sides of the polygon and the radii of curvature at the vertices. As a consequence, we obtain a discrete version of Ros’ inequality.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Using a notion of curvature at the vertices of a polygon, we prove an inequality involving the length of the sides of the polygon and the radii of curvature at the vertices. As a consequence, we obtain a discrete version of Ros’ inequality.
2014
J. Cufí, A. Reventós
EVOLUTES AND ISOPERIMETRIC DEFICIT IN TWO-DIMENSIONAL SPACES OF CONSTANT CURVATURE Journal Article
In: ARCHIVUM MATHEMATICUM (BRNO), vol. 50, pp. 219-236, 2014.
@article{1g,
title = {EVOLUTES AND ISOPERIMETRIC DEFICIT IN TWO-DIMENSIONAL SPACES OF CONSTANT CURVATURE},
author = {J. Cufí, A. Reventós},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/Versio-archivum-1.pdf},
year = {2014},
date = {2014-10-01},
journal = {ARCHIVUM MATHEMATICUM (BRNO)},
volume = {50},
pages = {219-236},
abstract = {We relate the total curvature and the isoperimetric deficit of a curve γ in a two-dimensional space of constant curvature with the area enclosed by the evolute of γ. We provide also a Gauss-Bonnet theorem for a special class of evolutes.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We relate the total curvature and the isoperimetric deficit of a curve γ in a two-dimensional space of constant curvature with the area enclosed by the evolute of γ. We provide also a Gauss-Bonnet theorem for a special class of evolutes.
Papers on Real and Complex Analysis
2018
J. Cufí; J. Verdera
Differentiability properties of Riesz potentials of finite measures and non-doubling Calderón–Zygmund theory Journal Article
In: Annali della Scuola Normale di Pisa, vol. XVIII, no. 3, pp. 1081-1123, 2018.
@article{1c,
title = {Differentiability properties of Riesz potentials of finite measures and non-doubling Calderón–Zygmund theory},
author = {J. Cufí and J. Verdera
},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/Differentiability030417.pdf},
year = {2018},
date = {2018-10-11},
journal = {Annali della Scuola Normale di Pisa},
volume = {XVIII},
number = {3},
pages = {1081-1123},
abstract = {We study differentiability properties of the Riesz potential, with kernel of homogeneity 2 − d in $R^d$, d ≥ 3, of a finite Borel measure. In the plane we consider the logarithmic potential of a finite Borel measure. We introduce a notion of differentiability in the capacity sense, where capacity is Newtonian capacity in dimension d ≥ 3 and Wiener capacity in the plane. We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies Lp differentiability in the Calderón–Zygmund sense for 1 ≤ p < d/d − 2. If d ≥ 3, we show 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. The result is sharp and depends on deep results in non-doubling Calderón–Zygmund theory. In the plane the situation is different. Surprisingly 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. Finally we obtain, for d ≥ 3, results on Peano second order differentiability in the sense of capacity with exceptional sets of zero Lebesgue measure.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study differentiability properties of the Riesz potential, with kernel of homogeneity 2 − d in $R^d$, d ≥ 3, of a finite Borel measure. In the plane we consider the logarithmic potential of a finite Borel measure. We introduce a notion of differentiability in the capacity sense, where capacity is Newtonian capacity in dimension d ≥ 3 and Wiener capacity in the plane. We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies Lp differentiability in the Calderón–Zygmund sense for 1 ≤ p < d/d − 2. If d ≥ 3, we show 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. The result is sharp and depends on deep results in non-doubling Calderón–Zygmund theory. In the plane the situation is different. Surprisingly 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. Finally we obtain, for d ≥ 3, results on Peano second order differentiability in the sense of capacity with exceptional sets of zero Lebesgue measure.
J. Cufí, X. Tolsa; J. Verdera
About the Jones–Wolff Theorem on the Hausdorff dimension of harmonic measure Journal Article
In: arXiv:1809.08026 , 2018.
@article{1,
title = {About the Jones–Wolff Theorem on the Hausdorff dimension of harmonic measure},
author = {J. Cufí, X. Tolsa and J. Verdera},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/JonesWolffTheo.pdf},
year = {2018},
date = {2018-10-11},
journal = {arXiv:1809.08026 },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
2017
J. Cufí; A. Nicolau; A. Seeger; J. Verdera
ON SQUARE FUNCTIONS WITH INDEPENDENT INCREMENTS AND SOBOLEV SPACES ON THE LINE Journal Article
In: Annali di Matematica Pura ed Applicata, vol. 197, pp. 905-940, 2017.
@article{1b,
title = {ON SQUARE FUNCTIONS WITH INDEPENDENT INCREMENTS AND SOBOLEV SPACES ON THE LINE},
author = {J. Cufí and A. Nicolau and A. Seeger and J. Verdera},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/cnsv-feb20.pdf},
year = {2017},
date = {2017-03-01},
journal = {Annali di Matematica Pura ed Applicata},
volume = {197},
pages = {905-940},
abstract = {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 $H_α^1$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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 $H_α^1$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.
2015
J. Cufí, J. Verdera
A general form of Green Formula and Cauchy Integral Theorem Journal Article
In: Proceedings of the American Mathematical Society, vol. 143, no. 5, pp. 2091-2102, 2015.
@article{1d,
title = {A general form of Green Formula and Cauchy Integral Theorem},
author = {J. Cufí, J. Verdera},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/Green_Cauchy.pdf},
year = {2015},
date = {2015-10-01},
journal = {Proceedings of the American Mathematical Society},
volume = {143},
number = {5},
pages = {2091-2102},
abstract = {We prove a general form of Green Formula and Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane. We use Vitushkin’s localization of singularities method and a decomposition of a rectifiable curve in terms of a sequence of Jordan rectifiable sub-curves due to Carmona and Cufí.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We prove a general form of Green Formula and Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane. We use Vitushkin’s localization of singularities method and a decomposition of a rectifiable curve in terms of a sequence of Jordan rectifiable sub-curves due to Carmona and Cufí.
J. Bruna, J. Cufí, H. Führ; M. Miró
Characterizing Abelian Admissible Groups Journal Article
In: Journal of Geometric Analysis , vol. 25, pp. 1045-1074, 2015.
@article{1i,
title = {Characterizing Abelian Admissible Groups},
author = {J. Bruna, J. Cufí, H. Führ and M. Miró},
year = {2015},
date = {2015-10-01},
journal = {Journal of Geometric Analysis },
volume = {25},
pages = {1045-1074},
abstract = {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. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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.
2013
J.J. Carmona, J. Cufí
The Calculation of the L^2-norm of the Index of a plane curve and related formulas Journal Article
In: Journal d'Analyse Mathématique, vol. 120, pp. 225-253, 2013.
@article{1j,
title = {The Calculation of the L^2-norm of the Index of a plane curve and related formulas},
author = {J.J. Carmona, J. Cufí},
url = {http://mat.uab.cat/web/jcufi/wp-content/uploads/sites/12/2018/10/PreprintCarmonaCufí.pdf},
year = {2013},
date = {2013-10-01},
journal = {Journal d'Analyse Mathématique},
volume = {120},
pages = {225-253},
abstract = {In this paper we provide formulae to calculate the $L^2$-norm of the index function of a rectifiable closed curve in the complex plane. Some applications to isoperimetric inequalities are given. The main tool used is the decomposition of any rectifiable closed curve in a sequence of Jordan curves plus some curves with null index functions and an exceptional set.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we provide formulae to calculate the $L^2$-norm of the index function of a rectifiable closed curve in the complex plane. Some applications to isoperimetric inequalities are given. The main tool used is the decomposition of any rectifiable closed curve in a sequence of Jordan curves plus some curves with null index functions and an exceptional set.