Let $X_1,\cdots,X_n$ be i.i.d drawn from uniform distribution on $[0,1]^d$, and let $$\eta_n:=\inf\{\eta\in (0,1):[\eta,1-\eta]^d \subset \mathrm{conv}\{X_1,\cdots,X_n\}\}.$$ Then it is easy to show that $n^{1/d}\eta_n=O_p(1)$. My question is, is it possible for the rate $n^{1/d}$ to climb up to $n$ (up to log factor)?
I also wish to run some numerical experiment to get a sense how fast $\eta_n$ converges. The main question would be, can we efficiently check, for a given point $x \in [0,1]^d$, is it in the convex hull generated by $n$ points $x_1,\cdots,x_n$?