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Metric entropy of convex hulls in Hilbert spaces

Wenbo Li, Werner Linde (2000)

Studia Mathematica

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 , . . . , | | t j | | ≤ 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 ( a j ) j = 1 ∞ .

Metric Entropy of Homogeneous Spaces

Stanisław Szarek (1998)

Banach Center Publications

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

Metrically convex functions in normed spaces

Stanisław Kryński (1993)

Studia Mathematica

Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.

Minimal pairs of bounded closed convex sets

J. Grzybowski, R. Urbański (1997)

Studia Mathematica

The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.

Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski-Rådström-Hörmander spaces

Jerzy Grzybowski, Diethard Pallaschke, Ryszard Urbański (2009)

Banach Center Publications

The theory of minimal pairs of bounded closed convex sets was treated extensively in the book authored by D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets. In the present paper we summarize the known results, generalize some of them and add new ones.

Minimal pairs of compact convex sets

Diethard Pallaschke, Ryszard Urbański (2004)

Banach Center Publications

Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and superdifferentials of a quasidifferentiable function (see Dem86). Since the sub- and superdifferentials are not uniquely determined, minimal representations are of special importance. In this paper we give a survey on some recent results on minimal pairs of closed bounded convex sets in a topological vector space (see PALURB). Particular attention is paid to the problem of characterizing minimal representatives...

Minimality in asymmetry classes

Michał Wiernowolski (1997)

Studia Mathematica

We examine minimality in asymmetry classes of convex compact sets with respect to inclusion. We prove that each class has a minimal element. Moreover, we show there is a connection between asymmetry classes and the Rådström-Hörmander lattice. This is used to present an alternative solution to the problem of minimality posed by G. Ewald and G. C. Shephard in [4].

Minimality of toric arrangements

Giacomo d'Antonio, Emanuele Delucchi (2015)

Journal of the European Mathematical Society

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.

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