{"id":1174694,"date":"2026-06-04T12:26:54","date_gmt":"2026-06-04T19:26:54","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/optimal-vector-balancing-for-zonotopes\/"},"modified":"2026-10-02T15:00:09","modified_gmt":"2026-10-02T22:00:09","slug":"optimal-vector-balancing-for-zonotopes","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/optimal-vector-balancing-for-zonotopes\/","title":{"rendered":"Optimal Vector Balancing for Zonotopes"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">A zonotope is a linear image of the cube <math><mrow><mo stretchy=\"false\">[<\/mo><mo>&#8211;<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mi>m<\/mi><\/mrow><\/math> for some <math><mrow><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><mi>m<\/mi><mi>a<\/mi><mi>t<\/mi><mi>h<\/mi><mi>b<\/mi><mi>b<\/mi><mi>N<\/mi><\/mrow><\/math>. We show that there is a universal constant <math><mi>C<\/mi><\/math> such that, for every zonotope <math><mrow><mi>Z<\/mi><mi>s<\/mi><mi>u<\/mi><mi>b<\/mi><mi>s<\/mi><mi>e<\/mi><mi>t<\/mi><mi>m<\/mi><mi>a<\/mi><mi>t<\/mi><mi>h<\/mi><mi>b<\/mi><mi>b<\/mi><msup><mi>R<\/mi><mi>d<\/mi><\/msup><\/mrow><\/math> and vectors <math><mrow><msub><mi>v<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>d<\/mi><mi>o<\/mi><mi>t<\/mi><mi>s<\/mi><mo>,<\/mo><msub><mi>v<\/mi><mi>n<\/mi><\/msub><mi>i<\/mi><mi>n<\/mi><mi>Z<\/mi><\/mrow><\/math>, there are signs <math><mrow><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mi>d<\/mi><mi>o<\/mi><mi>t<\/mi><mi>s<\/mi><mo>,<\/mo><msub><mi>x<\/mi><mi>n<\/mi><\/msub><mi>i<\/mi><mi>n<\/mi><mrow><mrow><mo>&#8211;<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/math> with [ sum_{i=1}^n x_i v_i in Csqrt d, Z. ] This resolves a 2002 question of Schechtman and generalizes Spencer&#8217;s six standard deviations theorem, which corresponds to the case <math><mrow><mi>Z<\/mi><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mo>&#8211;<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mi>d<\/mi><\/mrow><\/math>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A zonotope is a linear image of the cube [&#8211;1,1]m for some minmathbbN. We show that there is a universal constant C such that, for every zonotope ZsubsetmathbbRd and vectors v1,dots,vninZ, there are signs x1,dots,xnin&#8211;1,1 with [ sum_{i=1}^n x_i v_i in Csqrt d, Z. ] This resolves a 2002 question of Schechtman and generalizes Spencer&#8217;s [&hellip;]<\/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":[{"type":"user_nicename","value":"Victor 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