  The Shapley value is a common tool in game theory to evaluate the importance
of a player in a cooperative setting. In a geometric context, it provides a way
to measure the contribution of a geometric object in a set towards some
function on the set. Recently, Cabello and Chan (SoCG 2019) presented
algorithms for computing Shapley values for a number of functions for point
sets in the plane. More formally, a coalition game consists of a set of players
$N$ and a characteristic function $v: 2^N \to \mathbb{R}$ with $v(\emptyset) =
0$. Let $\pi$ be a uniformly random permutation of $N$, and $P_N(\pi, i)$ be
the set of players in $N$ that appear before player $i$ in the permutation
$\pi$. The Shapley value of the game is defined to be $\phi(i) =
\mathbb{E}_\pi[v(P_N(\pi, i) \cup \{i\}) - v(P_N(\pi, i))]$. More intuitively,
the Shapley value represents the impact of player $i$'s appearance over all
insertion orders. We present an algorithm to compute Shapley values in 3-D,
where we treat points as players and use the mean width of the convex hull as
the characteristic function. Our algorithm runs in $O(n^3\log^2{n})$ time and
$O(n)$ space. Our approach is based on a new data structure for a variant of
the dynamic convolution problem $(u, v, p)$, where we want to answer $u\cdot v$
dynamically. Our data structure supports updating $u$ at position $p$,
incrementing and decrementing $p$ and rotating $v$ by $1$. We present a data
structure that supports $n$ operations in $O(n\log^2{n})$ time and $O(n)$
space. Moreover, the same approach can be used to compute the Shapley values
for the mean volume of the convex hull projection onto a uniformly random $(d -
2)$-subspace in $O(n^d\log^2{n})$ time and $O(n)$ space for a point set in
$d$-dimensional space ($d \geq 3$).
