Permit me to revisit an earlier unresolved MO question, "Chord arrangement that avoids confining small or large disks" with a (very!) specific version, inspired by radiation therapy. The main idea is to minimize the size of tumor (a ball) that could be missed by radiation probes. More specifically,

Let $S$ be a unit sphere in $\mathbb{R}^d$. You are permitted to slice through $S$ with $n=d{+}1$ planes (of radiation), each $(d{-}1)$-dimensional flats. Is it the case that, in order to minimize the size of any missed(unradiated) ball within $S$, one should arrange the flats to determine a regular simplex, whose insphere has radius $r = \frac{1}{3}$?

I feel this should be a theorem for $n=d{+}1$ but (perhaps!)
very difficult to fathom for $n>d{+}1$.