Consider $\mathbb{R}^d$ under a general norm (not necessarily Euclidean). Consider two fixed concentric balls centered at the origin $B(0, R_1)$ and $B(0, R_1/2)$. Let $x$ be an arbitrary point with $|x| > R_1 > 0$ and $N_x$ be the $(d-1)$-hyperplane containing $x$ that is perpendicular to the line through $x$ and the origin. $C'_x$ is the convex hull of $\left\{ x\right\} \cup B(0,R_1/2)$.
Let $B$ denote some ball $B(x,r)$ with $r > 0$ centered at $x$. It is stated that $N$ is the "hyperplane" parallel to $N_x$ that contains $ \partial B \cap C'_x$.
How, in general, can it be true that such an $N$ exists? Taking the particular case of the euclidean norm on $\mathbb{R}^3$ as an illustration, it is not true that $\partial B \cap C'_x$ is contained in a $(d-1)$-hyperplane.