2 problems

Exact linear bound conjecture. The lower bound on $P(k)$ in Theorem is tight. Namely,

Let $[Z_1,\ldots,Z_{d+1}]$ be a $d$-dimensional simplex in $\mathbb{R}^{d+\ell}$, where $\ell\in\mathbb{N}_0$, and let $Z_1,\ldots,Z_{d+1}$ be independent and identically distribut…