Monika Ludwig

Ellipsoids and matrix-valued valuations

A classification is obtained of Borel measurable, GL(n) covariant, symmetric matrix valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by Lutwak, Yang, and Zhang. A classical concept from mechanics is the Legendre ellipsoid or ellipsoid of inertia Γ2K associated with a convex body K ⊂ R. It can be defined as the unique ellipsoid centered at the center of mass of K such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of K. If we fix a scalar product x · y for x, y ∈ R, Γ2K can be defined by the moment matrix M2(K) of K. This is the n × n matrix with coefficients ∫

Projection bodies and valuations

Let Π be the projection operator, which maps every polytope to its projection body. It is well known that Π maps the set of polytopes, Pn, in Rn into Pn, that it is a valuation, and that for every P∈Pn, ΠP is affinely associated to P. It is shown that these properties characterize the projection operator Π. This proves a conjecture by Lutwak.

A characterization of Lp intersection bodies

All GL(n) covariant Lp radial valuations on convex polytopes are classified for every p > 0. It is shown that for 0 < p < 1 there is a unique non-trivial such valuation with centrally symmetric images. This establishes a characterization of Lp intersection bodies. 2000 AMS subject classification: 52A20 (52B11, 52B45)

General affine surface areas

Abstract 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 ϕ affine surface areas are established.

Valuations on Sobolev spaces

All affinely covariant convex-body-valued valuations on the Sobolev space

Asymptotic approximation of smooth convex bodies by general polytopes

W^{1,1}({\Bbb{R}}^n)

Asymptotic approximation of convex curves

are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function

A characterization of affine length and asymptotic approximation of convex discs

f\in W^{1,1}({\Bbb{R}}^n)

Anisotropic fractional Sobolev norms

the unit ball of its optimal Sobolev norm.

VALUATIONS IN THE AFFINE GEOMETRY OF CONVEX BODIES

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we want to derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed. Let Pn and P(n) denote the set of polytopes with at most n vertices and n facets, respectively. For a convex body C, i.e., a compact convex set with non-empty interior, we are interested in the asymptotic behavior as n→∞ of