arXiv research

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

Paper solves capillary Orlicz-Minkowski problem with new inequalities.

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

The paper solves a specific Minkowski problem for capillary hypersurfaces.

Paper proves rigidity results for anisotropic capillary hypersurfaces.

Paper solves capillary Lp-Minkowski problem for p>1.

The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.

Proves conjectured capillary Blaschke-Santaló inequality for certain convex hypersurfaces.

New capillary Christoffel-Minkowski problem solved for half-space.

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known ex…

Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.

Study on convex capillary hypersurfaces with Lp curvature in half-space.

Proves existence of smooth convex solutions to capillary curvature equations.

Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.

The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.

Study inverse curvature flows for capillary hypersurfaces in a unit ball.

Study proves rigidity of capillary surfaces in curved 3D spaces.

Study of manifolds with specific curvature properties using capillary surfaces.

Paper solves inequalities for capillary hypersurfaces in half-spaces.

The paper proves inequalities for convex capillary hypersurfaces in a half-space.

The article proves inequalities for capillary hypersurfaces in hyperbolic space.

Paper introduces capillary Schwarz symmetrization in half-space.

Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.

The paper proves inequalities for hypersurfaces in a unit ball with specific boundary conditions.

The paper proves the existence of capillary geodesics on Riemannian 2-disks.

New Minkowski inequality for capillary surfaces in half-space.

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

Study shows how curved surfaces evolve smoothly to spherical shapes.

Study on minimizing singular capillary cones with stability and instability results.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

In this paper we prove that a capillary minimal surface outside the unit ball in $\mathbb {R}^3$ with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary minimal surface outside the unit ball with one end asymptotic to the end of the Ennep…

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundary surface co…

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. In Theorem 3, they provide an example of a capillary surface in a unit disk…

New index characterizes non-smooth Zoll convex bodies.

Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

The paper proves no multiple equichordal points exist in convex bodies.

Strongly convex bodies can be approximated by smooth ones.

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

Let $Σ$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $Σ$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial Σ$ is embedded for $n=2$, or if $\partialΣ$ is convex…

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

The paper proves a theorem linking convex body centroids and category theory.