The setting is a convex cone $C$ in $\mathbb{R}^d$ with the property that if you cut it with $S^d$ the volume of the cut is greater than or equal to $\operatorname{Vol}(S^d)/d+1$. That is, the volume of the cone (i.e. the volume of its cut with $S^d$) is greater than or equal to the volume of the cone from the standard partition from $\mathbb{R}^d$ into $d+1$ cones.
The cut of the boundary of $C$ and $S^d$ determines a $d-1$ hyperplane. It is stated now, that the distance from this hyperplane to the origin is less than or equal to $\frac{1}{2}$.
In dimension $d=2$, this fact is easy to confirm by simple triangle properties. I would be very grateful though for an aproach in arbitrary dimensions.
Edit: I realised that the distance should decrease with increasing dimension, so probably its a good approach to just show that, since for d=2 its confirmed
