\(L_{p}\) linear discrepancy of totally unimodular matrices (Q861031)

This note studies, for \(p\geq 1\), the \(L_p\) linear discrepancy of a totally unimodular matrix \(A\), \(\text{lindisc}_p (A)\). It is shown that for all natural numbers \(n\) there exists a totally unimodular matrix \(A\in\{0,1\}^{(n+1)\times n}\), satisfying \(\text{lindisc}_p(A)\geq c_p (1+o(1))\), where \[ c_p=\max_{a\in [0,1]} \left((1-a)a^p + a(1-a)^p\right)^{1/p} \] and where the \(o(1)\) term depends only on \(n\). Moreover, it is shown that \[ \frac{p}{p+1}\left(\frac{1}{p+1}\right)^{1/p} \leq c_p \leq \frac{p}{p+1}\left(\frac{1}{p+1}\right)^{1/p} \left(1+2^{-p+2}\right) \] for \(p\geq 3\) and that, if \(p\) is natural, there exist totally unimodular \((p+1)\times p\) matrices satisfying \[ \text{lindisc}_p > \frac{p}{p+1}\left(\frac{1}{p+1}\right)^{1/p}. \]