We introduce a robust conformal prediction method based on the half-mass radius $\dP(x):=\inf\{r>0:P(B(x,r))>1/2\}$. Its empirical version is exactly the distance from a point to its $\mathbf{k}$-th nearest neighbour, with $\mathbf{k}=\lfloor n/2\rfloor+1$. The resulting conformal sets retain finite-sample marginal validity and asymptotically target the robust population central set $\Qba:=\{x\in\R^d:\dP(x)\leβ_α\}$. We prove consistency in symmetric-difference probability, derive non-asymptotic exponential bounds and obtain the rate $O((\log n/n)^{1/2})$ under a local mass condition. Under the same local regularity on the uncontaminated law, we also establish quantitative $O(η)$ stability, uniformly over arbitrary contaminations of sufficiently small mass $η$, prove Hausdorff convergence of the empirical central sets, and provide computable ball-based representations using only pairwise distances.

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