Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces
Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$wh...
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| Autor*in: |
Rajala, Tapio [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Systematik: |
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| Anmerkung: |
© Springer-Verlag 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Monatshefte für Mathematik - Springer Vienna, 1948, 164(2010), 3 vom: 10. Dez., Seite 313-323 |
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| Übergeordnetes Werk: |
volume:164 ; year:2010 ; number:3 ; day:10 ; month:12 ; pages:313-323 |
| Links: |
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| DOI / URN: |
10.1007/s00605-010-0271-3 |
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| Katalog-ID: |
OLC2059510260 |
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| 520 | |a Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. | ||
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10.1007/s00605-010-0271-3 doi (DE-627)OLC2059510260 (DE-He213)s00605-010-0271-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Rajala, Tapio verfasserin aut Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. Hausdorff measure Packing measure Density Metric space Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 164(2010), 3 vom: 10. Dez., Seite 313-323 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:164 year:2010 number:3 day:10 month:12 pages:313-323 https://doi.org/10.1007/s00605-010-0271-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_267 GBV_ILN_2006 GBV_ILN_2015 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4266 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 164 2010 3 10 12 313-323 |
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10.1007/s00605-010-0271-3 doi (DE-627)OLC2059510260 (DE-He213)s00605-010-0271-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Rajala, Tapio verfasserin aut Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. Hausdorff measure Packing measure Density Metric space Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 164(2010), 3 vom: 10. Dez., Seite 313-323 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:164 year:2010 number:3 day:10 month:12 pages:313-323 https://doi.org/10.1007/s00605-010-0271-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_267 GBV_ILN_2006 GBV_ILN_2015 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4266 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 164 2010 3 10 12 313-323 |
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10.1007/s00605-010-0271-3 doi (DE-627)OLC2059510260 (DE-He213)s00605-010-0271-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Rajala, Tapio verfasserin aut Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. Hausdorff measure Packing measure Density Metric space Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 164(2010), 3 vom: 10. Dez., Seite 313-323 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:164 year:2010 number:3 day:10 month:12 pages:313-323 https://doi.org/10.1007/s00605-010-0271-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_267 GBV_ILN_2006 GBV_ILN_2015 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4266 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 164 2010 3 10 12 313-323 |
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10.1007/s00605-010-0271-3 doi (DE-627)OLC2059510260 (DE-He213)s00605-010-0271-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Rajala, Tapio verfasserin aut Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. Hausdorff measure Packing measure Density Metric space Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 164(2010), 3 vom: 10. Dez., Seite 313-323 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:164 year:2010 number:3 day:10 month:12 pages:313-323 https://doi.org/10.1007/s00605-010-0271-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_267 GBV_ILN_2006 GBV_ILN_2015 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4266 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 164 2010 3 10 12 313-323 |
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comparing the hausdorff and packing measures of sets of small dimension in metric spaces |
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Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces |
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Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. © Springer-Verlag 2010 |
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Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. © Springer-Verlag 2010 |
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Abstract We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets $${E\subset X}$$ with $${0 < \mathcal{H}^s(E) < \infty}$$. As an immediate consequence we improve the upper bound for the lower s-density of such sets in $${\mathbb{R}^n}$$. © Springer-Verlag 2010 |
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Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces |
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