The radii of sections of origin-symmetric convex bodies and their applications☆
Introduction
We present a geometric inequality in which bounds the quotient of two norms , and induced by the convex bodies and respectively. These bounds are expressed in terms of expectations of the functions and on with respect to the invariant normalized measure. Estimates of expectations allow us to offer a new method of evaluation of -widths of multiplier operators in such situations where known methods meet difficulties of a fundamental nature. In the case of Sobolev classes on the torus , Corollary 2 is well-known [37, p. 197]. In the case of multiplier operators on , , Theorem 2 follows from Corollary 1 and [22, Theorem 1]. Observe that Corollary 2 gives sharp lower bounds for on two-point homogeneous spaces if , , which are new in the case [1]. Corollary 2 gives new lower bounds for on compact homogeneous Riemannian manifolds which are not two-point homogeneous spaces if , . It is shown that these estimates are order sharp if . Incorrect proofs of the orders , , , in the case of two-point homogeneous spaces were presented by Brown and Dai [6]. The authors use a well-known method (see e.g. [33, p. 234], [36, p. 419]) which employs compactly supported functions [6, p. 420]. This method is essentially based on the fact that the derivatives of order preserve supports of infinitely differentiable functions. More precisely, to show embedding (4.11) [6] the authors failed to prove that the action of multiplier operators, which correspond to the fractional derivatives of orders , on compactly supported functions preserves their supports.
As an application of our results, we prove the existence of flat polynomials on compact homogeneous Riemannian manifolds. Important examples of such manifolds are given by real and complex Grassmannians, -toruses, Stiefel manifolds, complex spheres and two-point homogeneous spaces, i.e. real spheres, real, complex and quaternionic projective spaces and the Cayley elliptic plane. The method’s possibilities are not confined to the statements proved but can be used in studying more general problems. In the last section we show that the results we derive are new in the one-dimensional case, on the torus . Observe that various tools from asymptotic geometric analysis have been recently applied to study the problem of numerical integration on general domains [10], [11], [20]. The radius , defined by (8), of a random intersection of a centered ellipsoid with half exes with a random subspace , codim has been recently estimated in [8], [9]. It was shown that with overwhelming probability, where is an absolute constant.
Positive constants which enter into the estimates are mostly denoted by the letter . The same letter will be used to denote different universal constants in different parts of the article. For ease of notation we will write for two sequences, if , and , if , and some constants , and . denotes the integer part of .
Section snippets
Geometric inequality
Let be the canonical basis in , , and . Also, let be the Euclidean norm on , , be the unit sphere in , be the unit ball in and be the standard -dimensional volume of subsets in . The norm in is defined as usual, and . Fix a norm on and denote by the Banach space with the unit ball
Examples and applications
In this section we give various applications of Theorem 1. As a motivating example consider compact, connected, -dimensional Riemannian manifold , with metric. Let its metric tensor, its normalized volume element and its Laplace–Beltrami operator. In local coordinates , , Here, , and . It is well-known that is an elliptic, self adjoint, invariant under isometries, second-order operator. The
Acknowledgments
The authors wish to thank referees and Communicating Editor for valuable comments.
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Communicated by J. Prochno.