Abstract
On probability measures with a maximum of quantization dimension
Highlights
It is known that the quantization dimension of a probability measure on a metric compact space does not exceed the box-dimension of its support
The following question naturally arises about intermediate values of quantization dimensions
Let (X, ρ) be a metric compact with box-dimension dimB X = d. Is it true that for any a ∈ [0, d] there exists a probability measure μ with support supp(μ) = X for which the quantization dimension D(μ) is a? In this paper we consider a special case of this question concerning the existence of measures whose quantization dimension takes the largest possible value, which is equal to dimB X
Summary
It is known that the quantization dimension of a probability measure on a metric compact space does not exceed the box-dimension of its support. Пусть (X, ρ) — метрический компакт емкостной размерности dimB X = d. Что для любого a ∈ [0, d] существует вероятностная мера μ с носителем supp(μ) = X, для которой размерность квантования D(μ) равна a?
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