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Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, define

$$ \beta_\varepsilon(C) := \inf_{v \in B_d} \frac{\operatorname{vol}_d(C \cap (\varepsilon v + C))}{\operatorname{vol}_d(C)}, $$ where $v + C := \{v + c \mid c \in C\}$. Note that it always holds that $\beta_0(C) = 1$.

Question. For fixed $\operatorname{vol}_d(C)$ and $\varepsilon > 0$, what choices for $C$ maximize $\beta_\varepsilon(C)$?

My rough guess is that $\beta_\varepsilon$ is maximized for balls, but I'm not quite sure how to establish such a result. This is indeed the case when we require $C$ be convex, as established here https://mathoverflow.net/a/282941/78539 (thanks user J.G for reminding me!), with $\inf_v$ replace with $\mathbb E_v$.


Also, same question with $\beta_\varepsilon(C)$ replaced with (perhaps the more appriopriate quantity?) $\liminf_{\varepsilon \to 0^+} \dfrac{\beta_\varepsilon(C)-1}{\varepsilon}$.

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  • $\begingroup$ Something is wrong here. Clearly, the $\sup$ is $1$, attained at $v=0$ (if $vol_d(C)>0$). $\endgroup$ Feb 1, 2022 at 18:10
  • $\begingroup$ Indeed, typo. I meant "minimize ==> maximize", and "sup ==> inf". $\endgroup$
    – dohmatob
    Feb 1, 2022 at 18:30
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    $\begingroup$ Among convex sets, the answer seems to be yes for balls: the answer in mathoverflow.net/questions/282526/… shows that balls maximize your quantity over convex sets with the $\inf$ replaced by an expectation. For balls, the expectation is the same as the inf/sup, while for general sets the inf lower bounds the expectation, establishing the claim. It's hard for me to imagine anything fundamentally changes for nonconvex sets, but not sure. $\endgroup$ Feb 2, 2022 at 6:32
  • $\begingroup$ Indeed, the question is settled for convex sets, I forgot to state this, and I actually partipated in that thread a few years ago. I've updated the question to reflect this piecd of information. $\endgroup$
    – dohmatob
    Feb 2, 2022 at 12:37
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    $\begingroup$ @LSpice ok, sorry I misunderstood what you were saying. Thanks for the tip about linking comments (rather than entire post). Deleted my previous comment on this thread. $\endgroup$
    – dohmatob
    Feb 2, 2022 at 19:00

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