Dimension and geometry of sets defined by polynomial inequalities

Wallin, Hans; Wingren, Peter

doi:10.1016/0021-9045(92)90001-5

Cited by 8 publications

(4 citation statements)

References 10 publications

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“…On the other hand, L M I (1) implies the uniform perfectness. Let us also mention that uniform perfect sets in R N have positive Hausdorff dimension ( [25]). …”

Section: Definition 15mentioning

confidence: 99%

How to construct totally disconnected Markov sets?

Białas-Cież

Kosek

2010

Annali di Matematica

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We deduce a polynomial estimate on a compact planar set from a polynomial estimate on its circular projection, which enables us to prove Markov and Bernstein-Walsh type inequalities for certain sets. We construct -totally disconnected Markov sets that are scattered around zero in different directions; -a Markov set E ⊂ R such that neither E ∩ [0, +∞) nor E ∩ (−∞, 0] admit Markov's inequality; -a Markov set that is not uniformly perfect.Finally, we propose a construction based on a generalization of iterated function systems: a way of obtaining a big family of uniformly perfect sets.

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“…On the other hand, L M I (1) implies the uniform perfectness. Let us also mention that uniform perfect sets in R N have positive Hausdorff dimension ( [25]). …”

Section: Definition 15mentioning

confidence: 99%

How to construct totally disconnected Markov sets?

Białas-Cież

Kosek

2010

Annali di Matematica

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“…The Markov property defined by inequality (17) can be found in [3,8,20] Bos and Milman (A Geometric Interpretation and the Equality of Exponents in Markov and Gagliardo-Nirenberg (Sobolev) Type Inequalities for Singular Compact Domains (preprint)). If a set E ⊂ C has the LMP(m, κ) with constant c 1 , then c n in inequality (17) equals c 1 n κ and in the proof of Theorem 4.3 we can put a k = c k m+κ with some constant c independent of z 0 ∈ E, r ∈ (0, 1] and k ∈ {2, 3, ...}.…”

Section: Proposition 42 For Any Compact Setmentioning

confidence: 99%

Equivalence of the Local Markov Inequality and a Kolmogorov Type Inequality in the Complex Plane

Białas-Cież¹,

Eggink²

2012

Potential Anal

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“…for every x ∈ A and every algebraic polynomial p : R n → R n of total degree at most k [40]. This inequality is important in polynomial approximation in function spaces defined on A [25].…”

Section: Introductionmentioning

confidence: 99%

The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets

Hästö¹

2006

Journal of Applied Analysis

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The Apollonian metric is a generalization of the hyperbolic metric to arbitrary open sets in Euclidean spaces. In this article we show that the Apollonian metric is comparable to the jG metric in the set G if and only if its complement is unbounded and thick in the sense of Väisälä, Vuorinen and Wallin [Thick sets and quasisymmetric maps, Nagoya Math. J. 135 (1994), 121-148]. These conditions are also equivalent to the following: there exists L > 1 such that all Euclidean Lbilipschitz mappings are Apollonian bilipschitz with uniformly bounded constant.2000 Mathematics Subject Classification. Primary 30F45; Secondary 30C65.

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Dimension and geometry of sets defined by polynomial inequalities

Cited by 8 publications

References 10 publications

How to construct totally disconnected Markov sets?

How to construct totally disconnected Markov sets?

Equivalence of the Local Markov Inequality and a Kolmogorov Type Inequality in the Complex Plane

The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets

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