Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T=t_1,t_2,...$, $\left|\right|t_j\left|\right|\le a_j$, by functions of the $a_j$’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ${\left(a_j\right)}_{j=1}^{\infty }$.

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{\sigma }$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{\delta }$ set G ⊆ ℝ such that every weak contraction...

Let ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a smooth submanifold of the Hilbert space ${\mathcal{B}}_2\left(\mathcal{H}\right)$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set ${\mathcal{I}}_N$, which consists of all partial isometries of rank $N<\mathrm{\infty }$. Furthermore, ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a homogeneous space of ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}\times {\mathcal{U}}_{\left(\mathrm{\infty }\right)}$, where ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...

We study the problem of whether ${}_w\left(ⁿE\right)$, the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if ${}_w\left(ⁿE\right)$ is an M-ideal in (ⁿE), then ${}_w\left(ⁿE\right)$ coincides with ${}_{w0}\left(ⁿE\right)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if ${}_w\left(ⁿE\right){=}_{w0}\left(ⁿE\right)$ and (E) is an M-ideal in...

We study mild n times integrated C-existence families without the assumption of exponential boundedness. We present several equivalent conditions for these families. Hille-Yosida type necessary and sufficient conditions are given for the exponentially bounded case.

We give a lower bound for the minimal displacement characteristic in Hilbert spaces.

We say that a function from $X=C^L[0,1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape” to preserve....