$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$

Question

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 $\mathbb E_v$.

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