arXiv subjects

Nico Lombardi

Publications and source records attributed to Nico Lombardi.

5 recordsLinked to original sources

arXiv2024

Canal Classes and Cheeger Sets

Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.

math.MG↗

arXiv2020

Plane $R$-paths and their rectifiability property

A family of plane oriented continuous paths depending on a fixed real positive number $R$ is considered. For any point $x$ on the path, the previous points lie out of any circle of radius $R$ having at $x$ interior normal in a suitable tangent cone to the path at $x$. These paths are locally descent curves of a nested family sets of reach $R$. Avoiding any smoothness requirements, we get angle estimate and not intersection property. Afterwards we are able to estimate the lenght and detour of this curve.

math.DS↗

arXiv2019

Affine Poisson & Non-Poisson trace principles for $\dot{H}^{-1<-α\le-\frac12}(\R^{n-1})$

This note discovers not only an affine non-sharp-Poisson trace inequality but also its sharp-non-Poisson version for a Sobolev function with the fractional antiderivative.

math.AP↗

arXiv2017

Translation invariant valuations on quasi-concave functions

We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations.

math.MG↗

arXiv2015

Valuations on the space of quasi-concave functions

We characterize the valuations on the space of quasi-concave functions defined on the $N$-dimensional Euclidean space, that are rigid motion invariant and continuous with respect to a suitable topology. Among them we also provide a specific description of those which are additionally monotone.

math.MG↗