Let $S=\{P_i\}$ be a set of $k$ points in $\mathbb{R}^n$ and let $C(S) \subset \mathbb{R}^n$ be their convex hull. Let $Q$ be the centroid of the points $P_i$, and let $S'$ be the set of $k$ points obtained by replacing an arbitrary point in $S$ with $Q$.
If $C(S)$ is open and the point replaced is a boundary point of $C(S)$, prove that $\mathrm{vol}(C(S')) < \mathrm{vol}(C(S))$, unless $S$ was degenerate with $\mathrm{vol}(C(S)) = 0$.
Is there a constant $0 < \gamma < 1$ (possibly depending on $k,n$), such that $\mathrm{vol}(C(S')) < \gamma \times\mathrm{vol}(C(S))$, under the same conditions as above.