Sharp minimax behavior as contamination approaches the breakdown boundary $\epsilon\uparrow 1/2$

Let $d,N\in\mathbb N$, let $K\subseteq\mathbb R^d$ be a nonempty star-shaped set with respect to some center $c\in K$ (that is, for every $x\in K$ and $t\in[0,1]$, $c+t(x-c)\in K$; the origin-centered condition is the special case $c=0$ used in the cited paper), and let $\epsilon\inPrasadan & Neykov (2024).

The clean sample is $Y_1,\dots,Y_N$ with $Y_i\overset{i.i.d.}{\sim}\mathcal N(\mu,I_d)$, where $I_d$ is the $d\times d$ identity matrix. An adversary is allowed to choose a (possibly data-dependent and randomized) index set $\mathcal O\subseteq\{1,\dots,N\}$ with $|\mathcal O|\le \epsilon N$ and replace $\{Y_i:i\in\mathcal O\}$ by arbitrary vectors in $\mathbb R^d$, producing observed data $X_1,\dots,X_N$. An estimator is any measurable map $\hat\mu:(\mathbb R^d)^N\to\mathbb R^d$. The squared-error minimax risk is