arXiv research

Study of smooth convex bodies up to congruence.

New index characterizes non-smooth Zoll convex bodies.

Strongly convex bodies can be approximated by smooth ones.

Study on Santaló point for convex bodies in normed spaces.

We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimat…

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

Solves Alexandrov's problem for hyperbolic convex bodies.

Paper extends Green-Osher inequality for convex bodies at dilation position.

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

Sharp stability results for reverse isoperimetric inequalities in 2D.

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 …

New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitab…

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

Employing a centro-affine flow on smooth convex bodies, we generate new centro-affine differential invariants. One class of the newly defined invariants is the object of a sharp isoperimetric inequality, while other new inequalities on known centro-affine invariants are obtained as a byproduct of the flow's study. Furt…

A new proof shows almost every normal to a smooth convex body intersects at least 6 normals from different points.

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiar…

Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function of $K$. Let $K, K_0\subset \R^n$ be smooth convex bodies of class $C^2_+$, and l…

Polytopes in high dimensions have at least 2n+4 normals.

The paper extends the convolution operator to non-smooth valuations using geometric inequalities.

The paper studies curvature measures and volume-preserving flows on convex bodies.

Solves Christoffel problem for disk area measures on spheres.

The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

New geometric inequalities for convex bodies derived from Log-Brunn-Minkowski conjecture.

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

Researchers solve a specific case of the $L^p$ Christoffel-Minkowski problem for $1 < p < k+1$.

Convex functions and bodies can be approximated by smoother convex functions.

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale f…

The paper solves a new Minkowski problem involving convex bodies and curvature measures.

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…

We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex functions on $\R^d$ which can be approximated by real analytic (or just smoother) c…

The study proves a localized ellipsoid characterization for convex bodies and applies it to Finsler surfaces.

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

New proof of Alesker's Irreducibility Theorem using localization techniques.

We study the motion of smooth, strictly convex bodies in $\mathbb{R}^n$ expanding in the direction of their normal vector field with speed depending on Gauss curvature and support function.

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

Proves stability of cone-volume measure with nearly constant density.

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

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…

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…

Billiard trajectories and geodesics are closely related geometrically.

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 hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …

This paper continues the study of a class of compact convex hypersurfaces in Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise naturall…

Paper solves capillary Orlicz-Minkowski problem with new inequalities.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.