On the inner parallel body of a convex body

It is shown that the set of boundary points of a convex body at which there are no interior tangent balls of positive radius has zero surface area.     

Determination of the body force of a two-dimensional isotropic elastic body

Let Ω$\Omega$ represent a two-dimensional isotropic elastic body. We consider the problem of determining the body force F$F$ whose form φ(t)(_f1(x),_f2(x))$\phi \left(t\right)(f_1\left(x\right),f_2\left(x\right))$ with φ$\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov’s regularization and truncated integration, we construct a regularized solution from the data given inexactly and explicitly derive the error estimate.     

On Ziegler's body relations

Zusammenfassung Im Hinblick auf das Kriechen von isotropen Metallen bei gleichzeitiger Spannung und beträchtlichem Temperaturgradienten werden Folgerungen der Zieglerschen Theorie der Stoffgleichungen bei veränderlicher Temperatur hergeleitet. Ausgangspunkt bildet der Ansatz (4) für die Dissipationsleistung je Volumeinheit, wov _ij den Kriechgeschwindigkeitstensor,s _{i j} den Spannungsdeviator,q _k den Wärmefluss undp _k die entsprechende fiktive Kraft bedeuten, und wo wie gewöhnlich die Summationskonvention benutzt worden ist. Unter der Annahme, dass der Einfluss vonp _k klein ist, werden die entsprchenden Stoffgleichungen (10) hergeleitet, wobieF(F _e , ) eine im allgemeinen nicht-lineare Funktion der Effektivspannung (v. Misessche Invariante vons _{i j} ) und die Absoluttemperatur ist. Wenn auf einachsige Fälle spezialisiert wird, folgt hier das System (13), wobei die Stoffkonstanteh ₁ den hypothetischen Kupplungseffekt zwischen Spannung und Wärmefluss darstellt.     

On convex body chasing

A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setF _i is specified, and the player has to move to a point inF _i. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceF _i the cost to the player is within a constant (multiplicative) factor of the off-line cost (i.e., the least possible cost when allF _i are known in advance). We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces. The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line/off-line problems such as thek-server problem; we make some remarks on these more general problems.Joel Friedman wishes to acknowledge the National Science Foundation for supporting this research in part under a PYI Grant, CCR-8858788, and IBM for an IBM faculty development award. Nathan Linial's work was supported by a grant from the Binational Science Foundation Israel/US and from the Israel Academy of Sciences.     

Affine convex body semigroups

In this paper we present a new class of semigroups called convex body semigroups which are generated by convex bodies of ? ^k . They generalize to arbitrary dimension the concept of proportionally modular numerical semigroups of Rosales et al. (J. Number Theory 103, 281–294, 2003). Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of ?² are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.     

The convex floating body

The convex floating body of a polytope is constructed and parametric variation of its volume and surface area functionals are illustrated through example.     

On the shape of a convex body with respect to its second projection body

We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\mathrm{\Pi }K)/V^{d-1}(K)$. Namely, we prove that if K is a 3-dimensional zonoid of volume 1, then its second projection body ${\mathrm{\Pi }}^{2}K$ is contained in 8K, while if K is any symmetric 3-dimensional convex body of volume 1, then ${\mathrm{\Pi }}^{2}K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperimetric problem and the best known lower bound up to date for $P(K)$ in 3 dimensions. As byproduct of our methods, we establish an almost optimal lower bound for high-dimensional bodies of revolution.     

Thermodynamic stability of continuous body

Necessary and sufficient conditions are given for a factorization that generalizes the special factorization. Applications to Wiener-Hopf equations and filtering are also presented.     

LOOMIS-WHITNEY＇S INEQUALITY ABOUT MIXED BODY

In this article, the authors establish two theorems for mixed body, which are the generalizations of the well-known Loomis-Whitney＇s inequality.     

The restricted three body problem revisited

We present a new computation of the Birkhoff normal form for the Hamiltonian of the restricted three body problem near the Lagrangian libration points. This leads to a new proof of the Lyapunov stability of these points.     

Inscribed simplexes of a convex body

This paper advances the theorem that for any smooth point M of the boundary of a convex body K Rⁿ there exists a nondegenerate simplex inscribed in K with a vertex at M that is similar to a given n-dimensional simplex. Similar problems are considered and unanswered questions are posed.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 47–49, 1992.     

Stability for a cap body inequality

There are inequalities of Favard and Minkowski in which equality holds for cap bodies in E ^d. By obtaining a lower bound for the first area measure, we show that if the Favard deficit is small for some body, then that body must be close, in terms of the Hausdorff metric, to a cap body. For d=3, the inequality of Minkowski is strengthened, and a weak stability result follows.Supported by National Science Foundation Grant DMS 8802674.     

Configurational forces in a micro-cracked body

A framework to model micro-cracked bodies based on the theory of continua with microstructure is presented: assuming a crack embedded in each material element, a field is considered characterising size and orientation of the crack, in addition to the usual placement in the Euclidian space. To deal with crack evolutions, we call upon configurational forces which, in the present case, occur at the level of the microstructure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal annulus of a convex body

It is my pleasant duty to thank Professor P. Gruber for his suggestions.