In Hengartner, W. and Theodorescu, R. (1973). Concentration Functions. Academic Press, New York. MR033144

they study the continuity properties and when the concentration function is zero even for the multivariate case in THEOREM 1.7.4..

The main idea goes through showing that the maximum is attained for each $t>0$ and so as $t\to 0$, the concentration function goes to zero.

The article ["Maximal Inequalities and Some Applications"][1] has many other references too.

Even with the extra supremum over the sphere, the same argument should work i.e. for each $t>0$ attaining attaining the supremum and using continuity of $X'$.


Some details. Let $f(x,t):=P(|X-x|\leq t)$. Since the set $S_{t}:=\{f(x,t):x\in \mathbb{R}^{n}\}$ is bounded, the supremum $L_{t}=\sup_{x}f(x,t)$ is finite i.e. there is a sequence  $\epsilon_{n}\to 0, x_{n}\in S_{t}$ such that

$$\epsilon_{n}-L_{t}\leq f(x_{n},t)\leq L_{t}.$$

So by boundedness of $y_{n}:=f(x_{n},t)\in [0,1]$, we can apply Bolzano-Weirstrass and continuity to get subsequence $x_{n_k}\to x^{*}_{t}$ and 

$$L_{t}\leq f(x^{*}_{t},t)\leq L_{t}\Rightarrow L_{t}=f(x^{*}_{t},t).$$

From here we simply use continuity in $t$-variable to get limit zero.

  [1]: https://www.motapa.de/open/schilling-kuehn-mmax-arxiv.pdf