{"schema_version":"dataset-v1","id":"feige-sharp-constant","version":1,"title":"The sharp constant by dimension","description":"For n independent nonnegative variables with means at most one, the smallest possible P(S < E S + 1) is (n/(n+1))ⁿ: 1/2, 4/9, 27/64, … It decreases to 1/e, Feige’s conjectured universal constant. The flat lines are the earlier proved universal lower bounds.","license":"CC0-1.0 (exact values)","problems":["feige-conjecture"],"statements":[{"problem":"feige-conjecture","statement":"feige-sharp-finite-unit-slack","role":"The sharp constant for each n: (n/(n+1))ⁿ."},{"problem":"feige-conjecture","statement":"feige-unit-slack","role":"The universal bound 1/e."},{"problem":"feige-conjecture","statement":"feige-conjecture-consequence","role":"Why 1/e is the limit of the dimension-wise constants."},{"problem":"feige-conjecture","statement":"feige-bound-0-1798","role":"0.1798 (2020)."},{"problem":"feige-conjecture","statement":"feige-bound-one-eighth","role":"1/8 (2010)."},{"problem":"feige-conjecture","statement":"feige-bound-one-thirteenth","role":"1/13 (2004)."}],"figure":{"family":"line-plot","title":"The sharp constant by dimension","caption":"The sharp constant for n variables, (n/(n+1))ⁿ, falls towards 1/e as n grows; earlier universal lower bounds sit well below it.","x_label":"number of variables n","y_label":"smallest P(S < E S + 1)","series":["sharp","limit","guo","hzz","feige"],"boundary":"Drawn for n ≤ 40. The values are exact; the theorems for every n and for 1/e are the linked statements. Garnett’s approximately 0.14 (2020) is not drawn: the source gives it only approximately.","statement":"feige-sharp-finite-unit-slack"},"provenance":{"method":"(n/(n+1))ⁿ computed as exact rationals for n = 1, …, 40 and checked to decrease; the flat lines are the cited constants 1/e, 0.1798, 1/8 and 1/13.","software":{"python":"3.12.15"},"precision":"Exact rationals, stored as floating point.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"sharp","label":"(n/(n+1))ⁿ, sharp","x":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"y":[0.5,0.4444444444444444,0.421875,0.4096,0.4018775720164609,0.396569456603966,0.39269590377807617,0.38974434312894585,0.387420489,0.38554328942953175,0.3839952305608768,0.3826967066770906,0.3815918725444634,0.3806403926722973,0.37981240581524567,0.37908533191793614,0.3784417801355261,0.3778681389925449,0.3773536025353077,0.3768894828730007,0.3764687154334921,0.3760854967156027,0.375735014906632,0.3754132467271024,0.37511680225396427,0.3748428050028718,0.37458879826339175,0.3743526712176019,0.3741326001327005,0.3739270011573595,0.3737344921355526,0.3735538614900617,0.37338404269432135,0.3732240931956915,0.37307317691013153,0.372930549601623,0.37279554660653674,0.3726675724755776,0.3725460921926982,0.37243062369780505]},{"key":"limit","label":"1/e, the limit","x":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"y":[0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233,0.36787944117144233]},{"key":"guo","label":"0.1798 (2020)","x":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"y":[0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798,0.1798]},{"key":"hzz","label":"1/8 (2010)","x":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"y":[0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125]},{"key":"feige","label":"1/13 (2004)","x":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"y":[0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693,0.07692307692307693]}],"marks":[]},"sha256":"8273cb9c4adbc6312f1d791892f439c7f2553bbf8c76254a08f77062d119e6c0"}