Compact sets of the complex plane having the K-property ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:47:19Z http://mathoverflow.net/feeds/question/68742 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68742/compact-sets-of-the-complex-plane-having-the-k-property Compact sets of the complex plane having the K-property ? Adrien Hardy 2011-06-24T15:22:16Z 2011-07-11T18:05:46Z <p>I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples.</p> <p>As it is classical to do in potential theory, denote for $\mu$ in $M_1(K)$, the set of probability measures on a compact set $K\subset \mathbb{C}$, its logarithmic energy by $$I(\mu)=\iint \log\frac{1}{|x-y|}d\mu(x)d\mu(y)$$ and define the capacity of a compact set $K\subset\mathbb{C}$ as $$Cap(K)=\exp\Big(-\inf_{\mu\in M_1(K)} I(\mu)\Big).$$ $K$ is said to satisfy the K-property at $z\in K$ if there exists $\rho_z > 0$ and $k_z> 0$ such that $$Cap(K\cap D(z,\rho))\geq \rho^{k_z}$$ for any $0&lt; \rho &lt; \rho_z$, where $D(z,\rho)$ stands for the disc centered at $z$ with radius $\rho$. We say that $K$ satisfies the K-property if it satisfies the K-property at every $z\in K$. </p> <p>One can show that segments, or circles, satisfy this K-property.</p> <p>Questions :</p> <ul> <li>Example of compact sets with positive capacity which do not have the K-property ? </li> <li>More generally, do you have references about K-property for compact sets ?</li> </ul> http://mathoverflow.net/questions/68742/compact-sets-of-the-complex-plane-having-the-k-property/68762#68762 Answer by Margaret Friedland for Compact sets of the complex plane having the K-property ? Margaret Friedland 2011-06-24T20:15:32Z 2011-06-25T19:19:12Z <p>I do not have a satisfactory answer to your question, just a pointer. In the paper:</p> <p>Białas-Cież, Leokadia Markov sets in ${\bf C}$ are not polar. Bull. Polish Acad. Sci. Math. 46 (1998), no. 1, 83–89</p> <p>for a compact subset $E$ of $\mathbb{C}$ which satisfies Markov inequality (i.e., certain estimate for derivatives of polynomials) a lower bound is proved for the capacity of $E$ in terms of the diameter of $E$ raised to the power 1/3. This is a global estimate, not a local one that you are asking about, but there may be a relation. The author of the paper may know more, so I suggest asking her anyway, whether you get any more specific answers on MO or not.</p> <p>BTW, the question could benefit from having an additional tag "Potential theory". </p>