Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{ R}^{ n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{ -n}\sum_{ x \in \left\{ -1,1\right\}^n} \|Ax\|_{ \ell^{ \infty}}$ is large. They correspond to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate. Such matrices can be seen as a collection of fair tests on a fair coin where at least one outcome is typically atypical. We prove that, as $n \rightarrow \infty$, the quantity can scale as $$ \max_{ A \in \mathbb{ R}^{ n \times n}} \frac{ 1}{ 2^{ n}}\sum_{ x \in \left\{ -1,1\right\}^n} \|Ax\|_{ \ell^{ \infty}} = (1+o(1)) \cdot \sqrt{ 2\log{ n}}.$$ We also present candidate maximizers up to dimension $n \leq 8$ which appear to be highly structured and have nice closed-form solutions.
Categories: math.FA math.CO