On the Aleksandrov-Fenchel inequality involving zonoids

The equality case in the general quadratic inequality V(K, L, K ₁, ..., K _n–2)² V(K, K, K ₁, ..., K _n–2) V(L, L, K ₁, ..., K _n–2) for mixed volumes is settled under the assumption that K and L are centrally symmetric and K ₁, ..., K _n–2 are zonoids. This result partly confirms a conjecture on the general case made in an earlier paper.     

A stability estimate for the Aleksandrov-Fenchel inequality,with an application to mean curvature

In the quadratic Aleksandrov-Fenchel inequality for mixed volumes, stated as inequality (1) below, whereC ₁, ...,C _n-2 are smooth convex bodies, equality holds only if the convex bodiesK andL are homothetic. Under stronger regularity assumptions onC ₁,...,C _n-2, a stability estimate is proved, expressing thatK andL are close to homothetic if equality is satisfied approximately. This is applied to estimate explicitly the deviation of a closed convex hypersurface with mean curvature close to one from a unit sphere.     

On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body

We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality which relates the quermassintegrals of a convex body to those of an arbitrary hyperplane projection of . A consequence is the following fact: for any convex body , for any -dimensional subspace of and any 0$">,

where denotes the Euclidean unit ball and denotes volume in the appropriate dimension.

     

Jackson-type inequality on the sphere

In a recent paper, I introduced new moduli of smoothness for functions on the sphere which did not use averages and, as a result, had some interesting properties. The direct, Jackson-type, estimate of the best approximation by spherical harmonics using the new moduli will be proved here. Equivalence with the appropriate K-functionals will be given. Relations with the moduli used earlier will be shown and used to prove new results for these moduli.     

On the stability of the polygonal isoperimetric inequality

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices.     

On the stability of infinite-dimensional linear inequality systems

The principal aim of this paper is to study the stability of the solution set mapping of a system composed by an arbitrary set of linear inequalities in an infinite-dimensional space. The unknowns space is assumed to be metrizable, which allows us to measure the size of any possible perturbation. Conditions guaranteeing the closedness, the lower semicontinuity and the upper semicontmuity of this mapping, at a particular nominal system, are given in the paper. The more significant differences with respect to the finite dimensional case, previously approached in the context of the so-called semi-infinite optimization, are illustrated by means of convenient examples.     

A Ricci inequality for hypersurfaces in the sphere

Let Mⁿ be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere In [9], B. Smyth proved that if Mⁿ, n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering of Mⁿ is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of with nonnegative sectional curvature.Received: 11 November 2003     

Jackson inequality for Banach spaces on the sphere

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.      

A CR Poincaré inequality on the complex sphere

The main purpose of this paper is to prove a CR Poincaré inequality with sharp exponent on the sphere in complex space. We use the complex tangential gradient on the sphere instead of the usual Laplace-Beltrami gradient on the sphere.     

On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P_n(μ) of degree n is stable to infinitesimal perturbations of every invariant set I_m with |m| ≥ n. For each zonal number m, I_m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998     

对偶Aleksandrov-Fenchel不等式的稳定性

多个几何体(主要是凸体(convex boby)和星体(star body))相似“偏差”的一个度量方法被引进,在此度量下,利用R~n中H(?)lder不等式的一个加强获得了另一类对偶Aleksandrov-Fenchel型不等式的稳定性版本(stability version)。     

The sharp Jackson inequality in the spaceL p on the sphere

The sharp Jackson inequality in the spaceL _p, 1≤p<2, on the unit Euclidean sphereS ⁿ⁻¹ ,n≥3, is proved. Forn=2, it was established by N. I. Chernykh. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 50–62, July, 1999.     

Power stability for the bieberbach inequality

One investigates the problem of the determination of those >0 for which, in the class S of functions f(z)= z+c₂z²+..., regular and univalent in the circle ¦Z¦<1, the Koebe function yields a strict local maximum of the moduli of the coefficients D_n () (n=2,3,...), defined by the expansionTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 58–64, 1983.     

On the spectral problem in the linear stability study of flows on a sphere

A normal mode instability study of a steady nondivergent flow on a rotating sphere is considered. A real-order derivative and family of the Hilbert spaces of smooth functions on the unit sphere are introduced, and some embedding theorems are given. It is shown that in a viscous fluid on a sphere, the operator linearized about a steady flow has a compact resolvent, that is, a discrete spectrum with the only possible accumulation point at infinity, and hence, the dimension of the unstable manifold of a steady flow is finite. Peculiarities of the operator spectrum in the case of an ideal flow on a rotating sphere are also considered. Finally, as examples, we consider the normal mode stability of polynomial (zonal) basic flows and discuss the role of the linear drag, turbulent diffusion and sphere rotation in the normal mode stability study.     

Finite and uniform stability of sphere coverings

By attaching cables to the centers of the balls and certain intersections of the boundaries of the balls of a ball covering ofE ^d with unit balls, we can associate to any ball covering a collection of cabled frameworks. It turns out that a finite subset of balls can be moved, maintaining the covering property, if and only if the corresponding finite subframework in one of the cabled frameworks is not rigid. As an application of this cabling technique we show that the thinnest cubic lattice sphere covering ofE ^d is not finitely stable. The first two authors were partially supported by the Hungarian National Science Foundation under Grant No. 326-0413.