The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

A solution to Hammer's X-ray reconstruction problem

We propose algorithms for reconstructing a planar convex body K from possibly noisy measurements of either its parallel X-rays taken in a fixed finite set of directions or its point X-rays taken at a fixed finite set of points, in known situations that guarantee a unique solution when the data is exact. The algorithms construct a convex polygon Pk whose X-rays approximate (in the least squares sense) k equally spaced noisy X-ray measurements in each of the directions or at each of the points.It is shown that these procedures are strongly consistent, meaning that, almost surely, Pk tends to K in the Hausdorff metric as k→∞. This solves, for the first time in the strongest sense, Hammer's X-ray problem published in 1963.     

Capacities, surface area, and radial sums

A dual capacitary Brunn-Minkowski inequality is established for the (n−1)-capacity of radial sums of star bodies in Rn. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in Rn, 1?p<n, proved by Borell, Colesanti, and Salani. When n?3, the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition. Note that when n=3, the (n−1)-capacity is the classical electrostatic capacity. A proof is also given of both the inequality and a (different) equality condition when n=2. The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense (n−1)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.     

Convex bodies of constant width and constant brightness

In 1926 Nakajima (= Matsumura) showed that any convex body in R³ with constant width, constant brightness, and boundary of class C² is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.     

On the convergence of type I Hermite–Padé approximants

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for L? affine surface areas are established.     

On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals

We investigate Baker’s bivariate distributions with fixed marginals which are based on order statistics, and find conditions under which the correlation converges to the maximum for Fréchet-Hoeffding upper bound as the sample size tends to infinity. The convergence rate of the correlation is also investigated for some specific cases.     

On the construction of highly symmetric tight frames and complex polytopes

Many “highly symmetric” configurations of vectors in C^d${\mathbb{C}}^d$, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n   vectors for C^d${\mathbb{C}}^d$ (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.     

Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0$\lambda >0$ in terms of inequalities for their derivatives on (0,∞)$(0,\infty )$. When λ=1$\lambda =1$, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Asymptotic method for Dougall's bilateral hypergeometric sums

By means of the modified Abel lemma on summation by parts, a recurrence relation for Dougall's bilateral -series is established with an extra natural number parameter m. Then the steepest descent method allows us to compute the limit for m→∞, which leads us surprisingly to a completely new proof of the celebrated bilateral -series identity due to Dougall (1907). The same approach applies also to the bilateral very well-poised -series identity [J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907) 114-132].     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

On a characterization of the support functions

Letf:S ^n–1 be a support function. Then, for everya , if the functionu f(u)–a, u has a negative minimum, then a unique argument exists for which this minimum is attained. It is shown that the converse holds true under some obvious restrictions onf. A perturbation theorem for the space (S ^n–1) is given as an application.     

Maximal Blaschke products

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari–Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

Asymmetric norms and optimal distance points in linear spaces

In this paper we analyze the existence of points of a subset S of a linear space X where the shortest distance to a point x of X with respect to an asymmetric norm q is attained (q-nearest points). Since the structure of an asymmetric norm do not provide in general uniqueness of such points—due to the fact that the separation properties in these spaces are in general weaker than in normed spaces—we develop a technique to find particular subsets of the set of q-nearest points—that we call optimal distance points—that are also optimal for the norm qs associated to the asymmetric norm.     

Integral geometry for the 1-norm

Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in Rⁿ${\mathbb{R}}^n$ equipped with the metric derived from the p  -norm. This has, in effect, been investigated intensively for 1<p<∞$1<p<\infty$, but not for p=1$p=1$. We show that integral geometry for the 1-norm bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p  . We prove a Hadwiger-type theorem for Rⁿ${\mathbb{R}}^n$ with the 1-norm, and analogues of the classical formulas of Steiner, Crofton and Kubota. We also prove principal and higher kinematic formulas. Each of these results is closely analogous to its Euclidean counterpart, yet the proofs are quite different.