A note on Borsuk’s problem in Minkowski spaces

In 1993, Kahn and Kalai famously constructed a sequence of finite sets in $d$-dimensional Euclidean spaces that cannot be partitioned into less than (1.203 $\dots$ + $o$(1))${}^{\sqrt{d}}$ parts of smaller diameter. Their method works not only for the Euclidean, but for all $l_p$-spaces as well. In this short note, we observe that the larger the value of $p$, the stronger this construction becomes.