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Comp."],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:p>We give a combinatorial proof of the result of Kahn, Kalai and Linial [16], which states that every balanced boolean function on the <jats:italic>n<\/jats:italic>-dimensional boolean cube has a variable with influence of at least <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306008340_inline1\">\n\t      <jats:alt-text>$\\Omega\\bigl(\\frac{\\log n}{n}\\bigr)$<\/jats:alt-text>\n\t    <\/jats:inline-graphic>. The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants.<\/jats:p>\n\t  <jats:p>We also state some conjectures about optimal constants.<\/jats:p>","DOI":"10.1017\/s0963548306008340","type":"journal-article","created":{"date-parts":[[2007,1,23]],"date-time":"2007-01-23T15:59:14Z","timestamp":1169567954000},"page":"693-712","source":"Crossref","is-referenced-by-count":20,"title":["Edge-Isoperimetric Inequalities and Influences"],"prefix":"10.1017","volume":"16","author":[{"given":"DVIR","family":"FALIK","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ALEX","family":"SAMORODNITSKY","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2007,9,1]]},"reference":[{"key":"S0963548306008340_manual_ref-7","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009809"},{"key":"S0963548306008340_manual_ref-16","first-page":"68","volume-title":"FOCS 1988","author":"Kahn","year":"1988"},{"key":"S0963548306008340_manual_ref-4","first-page":"287","volume-title":"Proc. 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