Lower Bounds for the Shadiness Constant of Finite-Dimensional Normed Spaces

Abstract: By the Hahn-Banach theorem, every normed space admits rank-one projections with operator norm one. However, this is not true for higher rank projections. Bosznay and Garay showed that for every $d \geq 3$ there exist $d$-dimensional normed spaces $X$ for which all projections of rank $k$, with $2 \leq k \leq d-1$, have norm larger than or equal to some constant $c>1$. We call the maximal such constant the shadiness constant of $X$. Although constructing such spaces is not difficult, few explicit estimates of their shadiness constants exist. We show how optimization techniques can provide provable lower bounds for these shadiness constants. As an application, we construct a $3$-dimensional normed space whose unit ball is a polytope with $12$ vertices, with shadiness constant at least $1.01$. Furthermore we show that there is no shady norm on $\mathbb{R}^3$ whose unit ball is a polytope with $10$ or fewer vertices, thereby confirming a conjecture by Bosznay and Garay.