![]() ![]() The projective and the injective tensor products of $0$-symmetric convex bodiesĪre continuous functions with respect to the Hausdorff distance. For example, if V is (n\ )\ 2-dimensional, then hyperspaces are (n-1\ )\ 1-dimensional subspaces (lines), and he claims that the (k\ )\ 0-dimensional subspaces (points) are the intersection of n-k 2-0 2 lines which should seem familiar. Higueras-Monta\~no Download PDF Abstract: In a preceding work it is determined when a centrally symmetric convex body The definition of hyperspace is an (n-1)-dimensionalsubspace. Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resource-bounded measure 0.Download a PDF of the paper titled A hyperspace of convex bodies arising from tensor norms, by Luisa F. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number β in (0,1) is shown to have dimension, the binary entropy of β. This gives a new characterization of algorithmic information and a new proof of Mayordomo’s recent theorem stating that the dimension of a sequence is the limit inferior of the average Kolmogorov complexity of its first n bits. The classical Hausdorff and packing dimensions work not only in Euclidean spaces, but in arbitrary metric spaces. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. The dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. It is shown that for every Δ02-computable real number α in there is a Δ02 sequence S such that dim(S)=α. Sequences that are random (in the sense of Martin-Löf) have dimension 1, while sequences that are decidable, Σ01, or Π01 have dimension 0. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x : y) and Mdim(x : y).Ī constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. ![]() Intuitively, these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. However, the extension properties of these hyperspaces in the asymptotic category remained unknown. (2)We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x
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