arXiv research

The study generalizes Santalo's formula and shows stability of trapping sets in Riemannian manifolds.

In this paper, we establish two Santaló type formulas for general Finsler manifolds. As applications, we derive a universal lower bound for the first eigenvalue of the nonlinear Laplacian, two Croke type isoperimetric inequalities, and a Yamaguch type finiteness theorem in Finser geometry.

Study on Santaló point for convex bodies in normed spaces.

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

Unified approach to Crofton and Hurwitz integral formulas for convex sets.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new…

We give a stability version of of the Blaschke-Santaló inequality in the plane.

Established a new inequality for convex bodies in high dimensions.

In this paper, using functional Steiner symmetrizations, we show that Meyer and Pajor's proof of the Blaschke-Santalo inequality can be extended to the functional setting.

Paper tackles Santaló's convex surface problem in hyperbolic 3-space.

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…

The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.

Study geodesic flows, billiards, and metrics on manifolds.

In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure ass…

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

The paper extends Santaló's ellipse measures to hitting probabilities for circle lattices.

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality …

In this paper, a new approach of defining Steiner symmetrization of coercive convex functions is proposed and some fundamental properties of the new Steiner symmetrization are proved. Further, using the new Steiner symmetrization, we give a different approach to prove a functional version of the Blaschke-Santalo inequa…

In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and prove some general properties and make them explicit them on simple surfaces, re…

This paper aims to develop basic theory for the dual Orlicz $L_φ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequalit…

Paper refines Talagrand inequality on Euclidean spaces.

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…

We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an (n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is uniquely…

The Funk metric connects billiards, projective geometry, and convex geometry.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

The paper extends inequalities for convex bodies to higher dimensions and various norms.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Derives an integral formula for G2-structures.

Derives integral formulae on weighted manifolds.

Paper proves a fixed point formula and applies it to a new proof of Harish-Chandra's character formula.

Unified formula for surfaces in Euclidean or Lorentzian 3-space.

Two tropical gluing formulas help calculate Gromov-Witten invariants.

Paper derives trace formula for magnetic Laplacian at zero energy.

The Gauss formula is extended to various Laplacians on submanifolds.

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…

Note on new cancellation formulas for manifolds.

The study identifies types of manifolds using variational formulas and integral-differential formulas.

The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …

Formula calculates volume of two-bridge knots.

Formula connects surgeries to Seiberg-Witten invariants.

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…

Introduces a universal Bochner formula for scalar curvature.

Extension formulae on almost complex manifolds studied with applications.

Calculates volumes of specific cone-manifolds using Schläfli formula.

Proves a formula for a special invariant of 4-manifolds.