Isoperimetric Problems in Quantitative form

The aim of this thesis is to study some open problems in the calculus of varations, such as the local minimality of the ball in Gauss space and an isoperimetric problem with a nonlocal term

Convergence of the volume preserving fractional mean curvature flow for convex sets

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \cite{CN} this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \cite{CSV2} imply the $C^{1+\alpha}$-regularity of the flow and then provide a regularity argument which improves this into $C^{2+\alpha}$-regularity of the flow. The regularity step from $C^{1+\alpha}$ into $C^{2+\alpha}$ does not rely on convexity and can probably be adopted to more general setting

Some isoperimetric inequalities involving the boundary momentum

The aim of this paper is twofold. In the first part we deal with a shape optimization problem of a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the bidimensional case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension. In the second part of this paper we focus on a weighted curvature integral. We prove a lower bound in any dimension and an upper bound in the planar case, which provides a stronger form of the classical isoperimetric inequality for convex sets.Comment: 20 page

A remark on a conjecture on the symmetric Gaussian Problem

In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a functional for certain value of the mass. We give a positive answer in dimension two while in higher dimension the situation is different. In fact, for small value of mass the ball centered at the origin is a local minimizer while for large values the ball is a maximizer among convex sets with uniform bound on the curvature

Some weighted isoperimetric inequalities in quantitative form

In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior of a negative power weight for the perimeter thus providing a complete picture of the isoperimetric problem in this context.Comment: We spotted a mistake in the proof of lemma 3.

A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter

A new method for evaluating the distribution of aggregate claims

In the present paper, we propose a method of practical utility for calculating the aggregate claims distribution in a discrete framework. It is an approximated method but unlike the other approximated methods proposed in the literature: the approximation concerns both the counting distribution and the convolution of the severity distributions; the approximation does not consist in truncating the original distribution up to a given number of terms nor in replacing it with another distribution or a more general function (but simply in considering only the significant numerical realizations and in neglecting the others); the resulting approximation of the aggregate claims distribution is lower than a prefixed maximum error (10(-6) in our applications). In particular, the probability distribution and also the first three moments are exact with the prefixed maximum error. The proposed method does not require special assumptions on the counting distribution nor the identical distribution of the severity random variables and it does not incur in underflow and overflow computational problems. It proves to be more flexible, easier and cheaper than the (exact and approximated) methods using recursion and Fast Fourier Transform. We show some applications using both a Poisson distribution and a Generalized Pareto mixture of Poisson distributions as counting distribution. In addition to the specific application proposed in this paper, the method can be applied in many other (life and nonlife) actuarial fields where the sum of discrete random variables and the calculation of compound distributions are involved. Besides, it can be extended in multivariate cases. (c) 2005 Elsevier Inc. All rights reserved