Given a set \(\Omega\subset\mathbb{R}^N\), and an amount of volume \(c\in (0, |\Omega|)\), the authors study the min-max problem \[ \inf\left\{\, \sup_{x\in\Omega}\{\, \mathrm{dist}(x; \omega)\,\}\,:\, |\omega|=c,\, \omega\subset\Omega\,\right\} \] Without any additional constraint on \(\omega\), the authors show that the problem is ill-posed as the infimum is zero and it is asymptotically attained by a sequence of disconnected sets with an increasing number of connected components. Assuming the convexity of the competitors \(\omega\), they show that there exist minima and that the function \[ f\colon (0, |\Omega|)\ni c\mapsto \inf\left\{\, \sup_{x\in\Omega} \mathrm{dist}\,(x; \omega)\,:\, |\omega|=c,\, \omega\subset\Omega\,\right\} \] is continuous and strictly decreasing (ref.first part of Theorem 1). One might naively expect the solution(s) \(\omega\) to inherit the symmetries of the ambient set \(\Omega\). The authors show this not to be the case, and that symmetry-breaking phenomena might happen already for the choice \(\Omega=[0,1]\times[0,1]\) (ref.Theorem 2).
It is worth mentioning, that they also show the problem to be equivalent to few other shape optimization problems (ref.second part of Theorem 1), which are more suitable to be treated numerically, and some qualitative and numerical results in the planar case appear in Sections 4 and 5.