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 $n>d{+}1$.

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    $\begingroup$ No. I think cake slicing strategies reduce it to a two dimensional problem. $\endgroup$ – The Masked Avenger Oct 6 '13 at 2:17
  • $\begingroup$ Of course, if you say ellipsoid instead of ball, then I don't know. $\endgroup$ – The Masked Avenger Oct 6 '13 at 2:52

As suggested by The Masked Avenger, this is certainly false for large $n$: using regularly spaced parallel planes one hits all spheres of radius larger than $1/n$. Note that this kind of strategy also works inside an ellipsoid, or any shape.

  • $\begingroup$ Agreed. However a different idea is needed if the tumors are not balls but just convex. $\endgroup$ – The Masked Avenger Oct 6 '13 at 16:28

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