Carsten Schütt

Entropy numbers of diagonal operators between symmetric Banach spaces

Abstract We give the exact order of the dyadic entropy numbers of the identities from l n p to l n r where p r . Weaker estimates can be found in [3, 4]. The crucial lemma is a combinatorial result from [5]. Then we consider (dyadic) entropy numbers of identities between finite-dimensional symmetric Banach spaces. We obtain a simple expression that gives the exact order up to some logarithmic factor. This allows us to generalize a theorem due to B. Carl [1] about diagonal operators. It turns out that the result still holds under much weaker assumptions on the spaces. More precisely, the assumptions are not so much concerned with the spaces themselves but (what seems to be intuitively clear) with the relation between the spaces.

Geometric and probabilistic estimates for entropy and approximation numbers of operators

Abstract It is an open problem whether the entropy numbers en ( T ) of continuous linear operators T : X → Y are essentially self-dual, i.e., e n (T) ∼ e n (T ∗ ) . We give a positive result in the case that X and Y ∗ are of type 2, using volume estimates. This generalizes a result of Carl (On Gelfand, Kolmogorov, and entropy numbers of operators acting between special Banach spaces, University of Jena, Jena, East Germany, 1983, preprint). Moreover, we derive bounds for the approximation numbers a n ( T ) of T by probabilistic averaging. The formulas are applied to determine the exact asymptotic order of the approximation numbers of the formal identity map between various sequence spaces as well as tensor product spaces. In the special case of l p n , the result was first proved by Gluskin ( Mat. Sb. 120 (1983) , 180–189. [Russian]) using a different method.

ON THE AFFINE SURFACE AREA

It is shown that at least two expressions that extend the definition of the affine surface area to all convex bodies coincide.

Projection constants of symmetric spaces and variants of Khintchine's inequality

Abstract The projection constants of the lpn-spaces for 1 ≦ p ≦ 2 satisfy with in the real case and in the complex case. Further, there is c < 1 such that the projection constant of any n-dimensional space Xn with 1-symmetric basis can be estimated by . The proofs of the results are based on averaging techniques over permutations and a variant of Khintchines inequality which states that

Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body

Let K be a convex body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gaus-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K

On the minimum of several random variables

\lim_{ N\to \infty} \frac{{\rm vol}_n (K ) -\mathbb{E} (f,N )}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}} = c_n\int_{\partial K}\frac{\kappa(x)^{\frac{1}{n-1}}}{f(x)^{\frac{2}{n-1}}} {\rm d}\mu_{\partial K}(x),

Lorentz spaces that are isomorphic to subspaces of

where c n is a constant depending on the dimension n only.

Homothetic floating bodies

It is proved that in order to study unconditional structures in tensor products of finite dimensional Banach spaces it is enough to consider a certain basis. This result is applied to spaces ofp-absolutely summing operators showing their “bad” structure.