A Localized Besicovitch-Federer Projection Theorem
The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every d-dimensional linear subspace. In fact, there exist maps whi...
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| creator | Pugh, Harrison |
| description | The classical Besicovitch-Federer projection theorem implies that the
d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible
d-unrectifiable part will strictly decrease under orthogonal projection onto
almost every d-dimensional linear subspace. In fact, there exist maps which are
arbitrarily close to the identity in the C^0 topology which have the same
property. A converse holds as well, yielding the following rectifiability
criterion: under mild assumptions, a set is rectifiable if and only if its
Hausdorff measure is lower semi-continuous under bounded Lipschitz
perturbations. |
| doi_str_mv | 10.48550/arxiv.1607.01758 |
| format | Article |
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d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible
d-unrectifiable part will strictly decrease under orthogonal projection onto
almost every d-dimensional linear subspace. In fact, there exist maps which are
arbitrarily close to the identity in the C^0 topology which have the same
property. A converse holds as well, yielding the following rectifiability
criterion: under mild assumptions, a set is rectifiable if and only if its
Hausdorff measure is lower semi-continuous under bounded Lipschitz
perturbations.</description><identifier>DOI: 10.48550/arxiv.1607.01758</identifier><language>eng</language><subject>Mathematics - Functional Analysis</subject><creationdate>2016-07</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1607.01758$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1607.01758$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Pugh, Harrison</creatorcontrib><title>A Localized Besicovitch-Federer Projection Theorem</title><description>The classical Besicovitch-Federer projection theorem implies that the
d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible
d-unrectifiable part will strictly decrease under orthogonal projection onto
almost every d-dimensional linear subspace. In fact, there exist maps which are
arbitrarily close to the identity in the C^0 topology which have the same
property. A converse holds as well, yielding the following rectifiability
criterion: under mild assumptions, a set is rectifiable if and only if its
Hausdorff measure is lower semi-continuous under bounded Lipschitz
perturbations.</description><subject>Mathematics - Functional Analysis</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzrFOwzAQgGEvDKjlAZiaF0i4c2L7MpaqBaRIMGSPLs5ZddXWyK2qwtMjCtO__fqUekSoGjIGnjhf46VCC64CdIbulV4WXfK8j98yFc9yij5d4tlvy41MkiUXHzntxJ9jOhb9VlKWw1zdBd6f5OG_M9Vv1v3qtezeX95Wy65k66g0gAKAYEKwYeKRuMGgqW2DG0PrSTyMwlqTkEdqma1oN9YgGCxhaOqZWvxtb-jhM8cD56_hFz_c8PUPxlU-_g</recordid><startdate>20160706</startdate><enddate>20160706</enddate><creator>Pugh, Harrison</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20160706</creationdate><title>A Localized Besicovitch-Federer Projection Theorem</title><author>Pugh, Harrison</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a678-501e00105ff6fdab8a41f2899f7bf9c8ec0bea228e8c189aa6e27b30e1f681f43</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Mathematics - Functional Analysis</topic><toplevel>online_resources</toplevel><creatorcontrib>Pugh, Harrison</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Pugh, Harrison</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Localized Besicovitch-Federer Projection Theorem</atitle><date>2016-07-06</date><risdate>2016</risdate><abstract>The classical Besicovitch-Federer projection theorem implies that the
d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible
d-unrectifiable part will strictly decrease under orthogonal projection onto
almost every d-dimensional linear subspace. In fact, there exist maps which are
arbitrarily close to the identity in the C^0 topology which have the same
property. A converse holds as well, yielding the following rectifiability
criterion: under mild assumptions, a set is rectifiable if and only if its
Hausdorff measure is lower semi-continuous under bounded Lipschitz
perturbations.</abstract><doi>10.48550/arxiv.1607.01758</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Functional Analysis |
| title | A Localized Besicovitch-Federer Projection Theorem |
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