Given $C$ the convex hull of the points $Y_1,\ldots,Y_K$ in $\mathbb{R}^p$. Call $d$, the diameter of $C$. Suppose without loss of generality that $\|Y_1-Y_2\|=d$. Take $Y_0=(Y_1+Y_2)/2$. I need an upper bound for $\max\limits_{c\in C}\|c-Y_0\|$ in terms of $d$.
I thought the problem in $\mathbb{R}^2$, and got that $\max\limits_{c\in C}\|c-Y_0\|\le\sqrt{3}d/{2}$. I'm not quite sure, I thought that the points $Y_j$ necessarily live in the intersection of the balls $B(Y_1,d)$ and $B(Y_2,d)$ and the greatest possible distance to $Y_0$ is achieved in the points belonging to the intersections of the respective circles and this is by Pythagoras $\sqrt{3}d/2$. I believe the same bound might work for higher dimenssions.
Thank you for helping