For two metric spaces X and Y , say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps {\u03d5\u03c4 : X \u2192 Y : \u03c4 > 0} such that for every x, y \u2208 X, dX(x, y) \u2265 \u03c4 =\u21d2 dY (\u03d5\u03c4 (x), \u03d5\u03c4 (y)) \u2265 k\u03d5\u03c4 kLip\u03c4 \/K , where k\u03d5\u03c4 kLip denotes the Lipschitz constant of \u03d5\u03c4 . We show that if a metric space X thresholdembeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien\u2019s theorem. For instance, a subset X \u2286 L1 threshold-embeds into Hilbert space if and only if X has Markov type 2.<\/p>\n","protected":false},"excerpt":{"rendered":"
For two metric spaces X and Y , say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps {\u03d5\u03c4 : X \u2192 Y : \u03c4 > 0} such that for every x, y \u2208 X, dX(x, y) \u2265 \u03c4 =\u21d2 dY (\u03d5\u03c4 (x), \u03d5\u03c4 (y)) […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University Library","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2013-09-21","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1208.6088","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-357383","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University Library","msr_edition":"","msr_affiliation":"","msr_published_date":"2013-09-21","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1208.6088","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"https:\/\/arxiv.org\/abs\/1208.6088","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1208.6088"}],"msr-author-ordering":[{"type":"text","value":"Jian Ding","user_id":0,"rest_url":false},{"type":"text","value":"James R. 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