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We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features…
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is…
In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane,…
Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with…
We study the least gradient problem in bounded regions with Lipschitz boundary in the plane. We provide a set of conditions for the existence of solutions in non-convex simply connected regions. We assume the boundary data is continuous and…
In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using…
In this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with…
We give an overview of the 2022 Computational Geometry Challenge targeting the problem Minimum Partition into Plane Subsets, which consists of partitioning a given set of line segments into a minimum number of non-crossing subsets.
We address the double bubble problem for the anisotropic Grushin perimeter $P_\alpha$, $\alpha\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the case of two equal volumes. We assume that the contact interface between the bubbles…
We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of…
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…
We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for…
We study the isoperimetric problem in H-type groups and Grushin spaces, emphasizing a relation between them. We prove existence, symmetry and regularity properties of isoperimetric sets, under a symmetry assumption that depends on the…
Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean…
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem…
We study a constrained minimum energy problem with an external field relative to the Riesz kernel of an arbitrary order for a generalized condenser with touching oppositely-charged plates. Conditions sufficient for the solvability of the…
In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The…
Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection,…