The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.     

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball Bd⊂Rd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to Bd in Banach–Mazur distance converge to Bd in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of Bd. We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.     

The spherical conjecture in Minkowski geometry

We show that the shapes of convex bodies containing m translates of a convex body K, so that their Minkowskian surface area is minimum, tends for growing m to a convex body L.Received: 7 January 2002     

On the number of disjoint increasing paths in the one-skeleton of a convex body leading to a given exposed face

It is shown that ifF is ann-dimensional exposed face of ad-dimensional convex body and iff is a linear functional whose maximum value on the body is attained over the whole ofF thenn+1 paths can be found in the one-skeleton of the body leading toF and disjoint except at their end-points. Further, such paths may be found having the property that along them, the value off strictly increases. It is further shown that unlessn=0 it may be impossible to findn+2 such paths.     

Generic properties in Euclidean kinematics

The motion of a rigid body in a Euclidean space E ⁿ is represented by a path in the Euclidean isometry group E(n). A normal form for elements of the Lie algebra of this group leads to a stratification of the algebra which is shown to be Whitney regular. Translating this along invariant vector fields give rise to a stratification of the jet bundles J ^k (R, E(n)) for k=1, 2 and, hence, via the transversality theorem, to generic properties of rigid body motions. The relation of these to the classical centrodes and axodes of motions is described, together with applications to planar 4-bar mechanisms and the dynamics of a rigid body.     

The mixed area of a convex body and its polar reciprocal

Half the vector sum of a convex body and its polar reciprocal with respect to a unit sphereE containsE. A consequence of this is: The mixed area of a plane convex body and its polar reciprocal with respect toE is minimized by circles concentric withE. This work was supported in part by a grant from the National Science Foundation, NSF-G 19838. The author is indebted to the referee for fruitful comments.     

On the smallest disk containing a planar reduced convex body

A convex body R of Euclidean space E ^d is said to be reduced if every convex body $ P \subset R $ different from R has thickness smaller than the thickness $ \Delta(R) $ of R. We prove that every planar reduced body R is contained in a disk of radius $ {1\over 2}\sqrt 2 \cdot \Delta(R) $. For $ d \geq 3 $, an analogous property is not true because we can construct reduced bodies of thickness 1 and of arbitrarily large diameter.     

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.     

The Petty Projection Inequality for Lp-Mixed Projection Bodies

Recently, Lutwak, Yang and Zhang posed the notion of Lp-projection body and established the Lp-analog of the Petty projection inequality. In this paper, the notion of Lp-mixed projection body is introduced--the Lp-projection body being a special case. The Petty projection inequality, as well as Lutwak＇s quermassintegrals （Lp-mixed quermassintegrals） extension of the Petty projection inequality, is established for Lp-mixed projection body.     

Rate of Approximation of Regular Convex Bodies by?Convex Algebraic Level Surfaces

We consider the problem of the approximation of regular convex bodies in ℝ ^d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ ^d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is \frac1n\frac{1}{n}. Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then \frac1n\frac{1}{n} is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C ²-smoothness the rate of approximation is O(\frac1n²)O(\frac{1}{n^{2}}).     

New Solvable Variants of the Goldfish Many‐Body Problem

A technique to manufacture solvable variants of the “goldfish” many‐body problem is introduced, and several many‐body problems yielded by it are identified and discussed, including cases featuring multiperiodic or isochronous dynamics.     

Small holding circles

A circle C holds a convex body K if C does not meet the interior of K and if there does not exist any euclidean displacement which moves C as far as desired from K, avoiding the interior of K. The purpose of this note is to explore how small can be a holding circle. In particular it is shown that the diameter of such a holding circle can be less than the width w of the body but is always greater than 2w/3.     

The affine image of a convex body of constant breadth

A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that ann-dimensional convex body,n≧3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane.     

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.     

Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body

The stability for all generic equilibria of the Lie–Poisson dynamics of the \mathfrakso(4)\mathfrak{so}(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of \mathfrakso(n)\mathfrak{so}(n) are equilibrium points for the rigid body dynamics. In the case of \mathfrakso(4)\mathfrak{so}(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in \mathfrakso(3)\mathfrak{so}(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.     

Containment and Circumscribing Simplices

The following containment theorem is presented: If K and L are convex bodies such that every simplex that contains L also contains some translate of K , then in fact the body L must contain a translate of the body K . One immediate consequence of this theorem is a strengthened version of Weil's mixed-volume characterization of containment. Received October 31, 1995, and in revised form February 28, 1996.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.     

Sections of the Difference Body

Let K be an n -dimensional convex body. Define the difference body by K-K= { x-y | x,y ∈ K }. We estimate the volume of the section of K-K by a linear subspace F via the maximal volume of sections of K parallel to F . We prove that for any m -dimensional subspace F there exists x ∈ \bf R ⁿ , such that for some absolute constant C . We show that for small dimensions of F this estimate is exact up to a multiplicative constant. Received May 6, 1998, and in revised form July 23, 1998.