arXiv research

Derives exact formula for Minkowski sum of ellipsoids in N-space.

Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.

Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other …

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

The paper proves inequalities for hyperbolic sets and curves.

The paper classifies maximal translation surfaces in Lorentz-Minkowski space.

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimens…

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.

Study proves conditions for translating solitons to be planar.

New examples of Calabi-Yau metrics on cones with irregular smooth links.

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to non degenerate surfaces in Lorentz-Minkowski space.

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

If $Γ$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(Γ+Γ)=\mathsf{co}(Γ),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in $\mathbb{R}^3$ with range $Γ$ such that $\frac{1}{2}(…

The paper proves and analyzes Minkowski inequalities for nearly spherical domains.

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and suffic…

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…

Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfe…

The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symme…

Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, $L_p$-Minkowski and $L_p$-Brunn-Minkowski inequalities f…

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.

Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.

Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.

We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space $V$, obtained by intertwining the product and the duality transform of S. Alesker, may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valu…

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…

Paper proves anisotropic Minkowski inequality and related inequalities.

Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

Paper solves capillary Orlicz-Minkowski problem with new inequalities.

Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.

New method solves generalized Minkowski problem for torsional rigidity.

Paper solves Minkowski problem for p-harmonic measures.

Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.

Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.

The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed…

Researchers solved the even $L^p$-Minkowski problem under curvature pinching.

Proves equality in Minkowski inequality for static, flat manifolds.

Solves a generalized dual Minkowski problem for specific values of q.

Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkows…

New flow method solves Christoffel-Minkowski problem.

This paper solves the dual Minkowski problem for q-torsional rigidity.

Extends Minkowski stability proof to minimal decay assumptions.

Paper estimates diameter for Minkowski problem solutions.

Study proves only origin-centered spheres solve certain curvature problems.

Solves $L_p$-Gaussian chord Minkowski problem using Gauss curvature flow.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.