78 citations · 146 across the 17 of their papers we have counts for
18 papers
A reverse isoperimetric inequality in three-dimensional space forms
Kostiantyn Drach, Gil Solanes, Kateryna Tatarko
A -convex body in a three-dimensional space form of constant curvature is a compact convex set whose boundary has normal curvatures bounded below b…
Projective transformations of convex bodies and volume product
Florent Balacheff, Gil Solanes, Kroum Tzanev
In this paper we study certain variational aspects of the volume product functional restricted to the space of small projective deformations of a fixed convex body. In doing so, we…
The Weyl tube theorem for Kähler manifolds
Andreas Bernig, Joseph H. G. Fu, Gil Solanes +1
As sharpened in terms of Alesker's theory of valuations on manifolds, a classic theorem of Weyl asserts that the coefficients of the tube polynomial of an isometrically embedded ri…
Crofton formulas in pseudo-Riemannian space forms
Andreas Bernig, Dmitry Faifman, Gil Solanes
Crofton formulas on simply-connected Riemannian space forms allow to compute the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners,…
The Santaló point for the Holmes-Thompson boundary area
Florent Balacheff, Gil Solanes, Kroum Tzanev
We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is , and in the case where unit bal…
Uniqueness of curvature measures in pseudo-Riemannian geometry
Andreas Bernig, Dmitry Faifman, Gil Solanes
The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that th…
Curvature Measures of Pseudo-Riemannian Manifolds
Andreas Bernig, Dmitry Faifman, Gil Solanes
The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric -tensors. More precisely,…
Integral geometry of exceptional spheres
Gil Solanes, Thomas Wannerer
The algebras of valuations on and invariant under the actions of and are shown to be isomorphic to the algebra of translation-invariant…
Dual curvature measures in hermitian integral geometry
Andreas Bernig, Joseph H. G. Fu, Gil Solanes
The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space of dual unitarily invariant curvatur…
Kinematic formulas on the quaternionic plane
Andreas Bernig, Gil Solanes
We introduce different bases for the vector space of -invariant, translation invariant continuous valuations on the quaternionic plane and determine a…
Regularized Riesz energies of submanifolds
Jun O'Hara, Gil Solanes
Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs…
Contact measures in isotropic spaces
Gil Solanes
We revisit the contact measures introduced by Firey, and further developed by Schneider and Teufel, from the perspective of the theory of valuations on manifolds. This reveals a li…
Classification of invariant valuations on the quaternionic plane
Andreas Bernig, Gil Solanes
We describe the orbit space of the action of the group on the real Grassmann manifolds in terms of certain quaternionic…
Integral geometry of complex space forms
Andreas Bernig, Joseph H. G. Fu, Gil Solanes
We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to…
The geometry of canal surfaces and the length of curves in de Sitter space
Rémi Langevin, Gil Solanes
We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in corresponden…
Möbius invariant energies and average linking with circles
Jun O'Hara, Gil Solanes
We define and study a Möbius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a Möbius version of Banchoff-Pohl's n…
The Gauss-Bonnet theorem and Crofton type formulas in complex space forms
Judit Abardia, Eduardo Gallego, Gil Solanes
We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian…
Total curvature of complete surfaces in hyperbolic space
Gil Solanes
We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour. The result is given in term…