arXiv subjects

Marco Longinetti

Publications and source records attributed to Marco Longinetti.

8 recordsLinked to original sources

arXiv2024

R-hulloid of the vertices of a tetrahedron

The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is the union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such a case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $\rho$ such that $V$ is a $\rho$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.

math.MG↗

arXiv2024

On the configurations of four spheres supporting the vertices of a tetrahedron

A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $R$ for which the set of vertices of $T$ is an $R$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $R$-bodies.

math.MG↗

arXiv2024

On the complements of union of open balls and related set classes

Let a $R$-body be a closed set, complement of union of open balls of radius $R$ in the Euclidean space. Properties generalizing similar ones for convex sets are proved for the family of $R$-bodies; properties for the family of sets supported by spheres of radius $R$ (extension of the supporting hyperplane to convex bodies) are investigated. Comparison of that family with the sets of reach $R$ and with the $R$-rolling sets are studied. New properties for the previous families are proved, by using the $R$-cones, generalization of the convex cones.

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↗

arXiv2016

On variational problems related to steepest descent curves and self dual convex sets on the sphere

Let $\mathcal{C}$ be the family of compact convex subsets $S$ of the hemisphere in $\rn$ with the property that $S$ contains its dual $S^*;$ let $u\in S^*$, and let $ Φ(S,u)=\frac{2}{ω_n}\int_{S}\ < θ, u \ > \,\, dσ(θ). $ The problem to study $ \inf \big\{Φ(S,u), S \in \mathcal{C}, \, u\in S^* \big\} $ is considered. It is proved that the minima of $ Φ$ are sets of constant width $ π/2 $ with $ u $ on their boundary. More can be said for $n=3$: the minimum set is a Reuleaux triangle on the sphere. The previous problem is related to the one to find the maximal length of steepest descent curves for quasi convex functions, satisfying suitable constraints. For $ n=2 $ let us refer to \cite{Manselli-Pucci}. Here quite different results are obtained for $ n\geq 3$.

math.OC↗

arXiv2015

Bounding regions to plane steepest descent curves of quasi convex families

Two dimensional steepest descent curves (SDC) for a quasi convex family are considered; the problem of their extensions (with constraints) outside of a convex body $K$ is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on $K$. These regions are bounded by arcs of involutes of the boundary of $K$ and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self contracting sets (with opposite orientation) are considered, necessary and/or sufficients conditions for them to be subsets of a SDC are proved.

math.CA↗

arXiv2013

On steepest descent curves for quasi convex families in $R^n$

A connected, linearly ordered path $\ga \subset R^n$ satisfying $$ x_1\prec x_2\prec x_3 \in \ga, and x_1 \prec x_2 \prec x_3 \Rightarrow |x_2 - x_1| \leq | x_3 - x_1|$$ is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these paths are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length's bounds are proved for them.

math.AP↗

arXiv2008

Convex Hulls of Orbits and Orientations of a Moving Protein Domain

We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in R^3 acting on the space of pairs of anisotropic symmetric 3\times 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution.

math.MG↗