55 papers
This paper investigates a novel notion of polytope rigidity where edge lengths and face planarity are preserved while face shapes may vary, proving that convex polytopes are generically rigid in dimension three and conjecturing this holds for all dimensions .
This paper investigates the horofunction boundaries of Carnot groups by proving that all horofunctions are piecewise-defined via Pansu derivatives and demonstrating that filiform Lie groups of dimension provide the first examples where the boundary dimension deviates from the expected .
This paper characterizes the conditions under which the band-unfolding of a nested prismatoid results in a nonoverlapping layout, demonstrating that the previously known counterexample is essentially the only obstruction to this method's success.
This paper establishes a generalized version of Reshetnyak's theorem for quasiregular values mapping from generalized -manifolds with controlled geometry to Euclidean space , extending prior results by Kangasniemi and Onninen from the Euclidean setting.
This paper establishes a pointwise geometric description of nonlinear Lebesgue spaces, including their length structure, curvature bounds, and speed definitions for absolutely continuous curves, by first proving a nonlinear Fubini–Lebesgue theorem that identifies L^p curves with mappings into the space of L^p curves.
This paper introduces the Elastic Time Warping algorithm, which solves the time series matching problem in arbitrary metric spaces using a Hellinger kernel stretching penalty with cubic computational complexity.
This paper investigates the intrinsic geometry of polyhedra by utilizing barycentric algebras to characterize coordinate systems, specifically presenting a coalgebra-based algorithm for computing chordal coordinates in convex polygons that naturally yields the Catalan number enumeration of their triangulations.
This paper establishes the quantitative stability of Kantorovich potentials and optimal transport maps on metric measure spaces with lower Ricci curvature bounds, confirming a recent conjecture by Kitagawa, Letrouit, and Mérigot through a novel proof using heat kernel-regularized -transforms that avoids reliance on linear structures or sectional curvature bounds.
This paper claims to derive a strictly closed-form, non-iterative formula for the partition function by utilizing the Simplicial Spectral Decomposition of partition polytopes within the framework of root systems and Ehrhart theory.
This paper establishes the global rigidity of hyperbolic inversive distance circle packings on the 2-sphere, and equivalently of Koebe polyhedra in the Beltrami-Klein model, under mild assumptions on vertex links, thereby generalizing previous rigidity results and the uniqueness part of the Koebe-Andreev-Thurston Theorem to cases where adjacent circles need not touch.
This paper confirms the Bader-Kropholler-Vankov conjecture by proving that homological filling functions over a ring with a discrete norm are quasi-isometry invariants for all groups of type , utilizing a novel technique of equipping free chain complexes with geometric structures to extend cellular constructions to the algebraic setting.
This survey article explores the universal decay rate for approximating smooth convex bodies by polytopes with faces, tracing the concept from classical circle approximations to modern results on random polytopes, projection-based distances, and open problems regarding sharp constants and dimension dependence.
This paper generalizes the Atiyah problem and the Atiyah–Sutcliffe conjectures by introducing the -amplitude function, a tensor-based geometric inequality defined on finite graphs that recovers the original conjectures when the graph is complete.
This paper defines a speed measure for locally bounded variation curves in metric spaces, characterizes their continuity and absolute continuity through this measure, identifies its Radon-Nikodým derivative as the metric speed, and establishes an extension of the Banach-Zaretsky theorem.
This paper establishes that convex-cocompact representations of finitely generated groups into the isometry group of infinite-dimensional hyperbolic space form an open set, enabling the construction of new surface group representations via bending that are distinct from those classified by Monod and Py.
This paper focuses on convex polygons in the plane by summarizing and comparing Gibbs coordinates (based on entropy maximization and exponential functions) with Wachspress coordinates (rational functions), identifying their points of agreement and divergence, and demonstrating how Gibbs coordinates for polygons with rational vertices can be interpreted as algebraic functions.
This paper characterizes the Legendre, Laplace, and identity transforms as the unique continuous, contravariant valuations with specific conjugation or duality properties on spaces of convex and log-concave functions.
This paper investigates the geometric relationship between hyperbolic, elliptic, and parabolic disks and their supporting half-distance bands within the Beltrami–Cayley–Klein model, aiming to define and quantify their "closeness" through precise approximations of area and circumference.
This paper proves a sharp reverse isoperimetric inequality in three-dimensional space forms of constant curvature, demonstrating that among all -convex bodies with a fixed surface area, the -convex lens uniquely minimizes volume, thereby confirming Borisenko's Conjecture for non-zero curvature cases.
This paper proves the validity of the strong version of the union of uniform closed balls conjecture in the plane.