A Further Characterization Of Ellipsoids

In this note we extend a characterization of ellipsoids given in [6, Theorem 3.1.2.7], which is related to the classical characterization of ellipsoids as the only ovaloids with constant r-th affine mean curvature.     

Logarithmic Norms and Nonlinear DAE Stability

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certain unbounded operators, i.e. partial differential algebraic equations. Perturbation bounds are obtained from differential inequalities for any given norm by using the relation between Dini derivatives and semi-inner products. Simple discretizations are also considered.     

Dynamics and Stability of Collective Action Norms

A set of computational experiments investigated a model of formal and informal control, showing how selective incentives to work for the collective good may paradoxically lead to enforcement of antisocial norms that oppose the collective good. In these conditions, the widely cited effects of selective incentives, group cohesiveness, and second-order free riding on collective action may be inverted. Mathematical analysis provides some certain bounds on the model's behavior and relaxes several restrictive assumptions used in the simulation research. This complementary view deepens our understanding of second order social control as a solution to problems of collective action.     

On Blaschke's extension of Bonnesen's inequality

W. Blaschke established a Bonnesen-style inequality for the relative inradius and circumradius of a planar convex bodyK with respect to another. We sharpen this inequality by considering the radii of the minimal convex annulus ofK.     

John and Loewner Ellipsoids

John’s ellipsoid criterion characterizes the unique ellipsoid of globally maximum volume contained in a given convex body C. In this article local and global maximum properties of the volume on the space of all ellipsoids in C are studied, where ultra maximality is a stronger version of maximality: the volume is nowhere stationary. The ellipsoids for which the volume is locally maximum, resp. locally ultra maximum are characterized. The global maximum is the only local maximum and for generic C it is an ultra maximum. The characterizations make use of notions originating from the geometric theory of positive quadratic forms. Part of these results generalize to the case where the ellipsoids are replaced by affine copies of a convex body D. In contrast to the ellipsoid case, there are convex bodies C and D, such that on the space of all affine images of D in C the volume has countably many local maxima. All results have dual counterparts. Extensions to the surface area and, more generally, to intrinsic volumes are mentioned.     

双旋转不变Radon变换的一个等价刻画

给出双旋转不变Radon变换的一个等价刻画.证明了双旋转不变Radon变换的双旋转不变性可以被■×■×[0,1]×■上的函数所刻画.     

新椭球的一些性质

首先讨论了当多胞形的新椭球是球的充要条件,然后对算子Gamma(-2)的性质进行了探讨,得出了一些相关的不等式．     

Stability estimates for the Radon transform with restricted data and applications

In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the L^p$L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.     

Minimum-Volume Enclosing Ellipsoids and Core Sets

We study the problem of computing a (1+ε)-approximation to the minimum-volume enclosing ellipsoid of a given point set . Based on a simple, initial volume approximation method, we propose a modification of the Khachiyan first-order algorithm. Our analysis leads to a slightly improved complexity bound of operations for . As a byproduct, our algorithm returns a core set with the property that the minimum-volume enclosing ellipsoid of provides a good approximation to that of . Furthermore, the size of depends on only the dimension d and ε, but not on the number of points n. In particular, our results imply that for .We thank the Associate Editor and an anonymous referee for handling our paper efficiently and for helpful comments and suggestions.This author was supported in part by NSF through Grant CCR-0098172.This author was supported in part by NSF through CAREER Grant DMI-0237415.     

Some Characterizations of Ellipsoids by Sections

We characterize ellipsoids among convex bodies in E^d looking at the sections parallel to two or three hyperplanes.     

On a generalization of Blaschke's Rolling Theorem and the smoothing of surfaces

A generalization of Blaschke's Rolling Theorem for not necessarily convex sets is proved that exhibits an intimate connection between a generalized notion of convexity, various concepts in mathematical morphology and image processing, and a certain smoothness condition. As a consequence a geometric characterization of Serra's regular model is obtained and various problems in image processing arisng from the smoothing of surfaces with Sternberg's rolling ball algorithm are addressed. Copyright © 1999 John Wiley & Sons, Ltd.     

Ellipsoids of maximal volume in convex bodies

The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterizing ellipsoids of maximal volume contained in convex bodies.     

Radon transformations and zonoids

The class of compact sets known as zonoids or Steiner's (compact) sets, i.e., compact sets that are positive linear combinations (possibly, continuous ones) of segments, are described in terms of the Radon transformation.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 254–260, February, 1996.This research was partially supported by the Russian Foundation for Basic Research.     

External Ellipsoidal Estimation of the Union of Two Concentric Ellipsoids and Its Applications

For an arbitrary set representable as the convex hull formed by the union of two concentric ellipsoids we propose a method to construct a family of external undominated ellipsoidal approximations and represent the estimated set as the intersection of all estimates from a given family. A sufficient condition of undominated guaranteed ellipsoidal approximation of a convex compactum is derived. A method is described that for certain classes of sets (such as the intersection of an ellipsoid or a cone with two halfspaces) constructs a family of internal undominated ellipsoidal approximations using the previous formulas for the external estimates of the union of concentric ellipsoids.     

Norms and perfect graphs

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods.In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium on Operations Research, Passau, 9.–11. 9. 1987.