An isodiametric problem of fractal dimension

Autor: Bo Tan, Xin-Rong Dai, Jun Luo, Wei-Hong He
Rok vydání: 2014
Předmět:
Zdroj: Geometriae Dedicata. 175:79-91
ISSN: 1572-9168
0046-5755
DOI: 10.1007/s10711-014-0030-z
Popis: For $$00\right\} =1$$ and that there is a convex compact set $$A$$ ( $$=A(\lambda )$$ ) with $$\frac{{\mathcal H}^s(A\cap F_\lambda )}{|A|^s}=1$$ . Such a convex compact set $$A$$ is called an “extremal set” of $$F_\lambda $$ with respect to $$s$$ -dimensional Hausdorff measure $${\mathcal H}^s$$ . When $$\lambda $$ is small, say $$\lambda \le \frac{1}{5}$$ , we further show that there exists an extremal set $$A$$ with $$|A|\ge \frac{2}{\sqrt{3}}$$ such that $${\mathcal H}^s(A\cap F_\lambda )={\mathcal H}^s(D_{|A|}\cap F_\lambda )$$ for $$D_{|A|}=\left\{ (x,y): \left( x-\frac{1}{2}\right) ^2+y^2 \le \frac{1}{4}|A|^2\right\} $$ . As an application, we can estimate the value of $${\mathcal H}^s(E_\lambda \!\times \![0,1])$$ to any pre-set error $$\epsilon $$ .
Databáze: OpenAIRE
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