Linear And Hereditary Discrepancy

Let A be a m n{matrix. Linear and hereditary discrepancy are deened as lindisc(A) := maxfminfkA(p ")k 1 j " 2 ff1; 1g n g jp 2 1; 1] n g herdisc(A) := maxfminfk(a ij) i2m];j2J "k 1 j " 2 ff1; 1g n g jJ n]g: We're investigating connections between the two concepts. Best results known are lindisc(A) 2(1 2 2 n) herdisc(A) for any matrix A and an example of an A satisfying lindisc(A) = 2(1 1 n+1) herdisc(A). We show lindisc(A) 2 1 2 blog 2 (m)cc1 herdisc(A) 2 1 1 2m herdisc(A) : Unless m 2 2 n 1 , this is better than the known results. We provide an example showing that our bound is almost the best possible (in terms of m).