  This paper presents the first systematic study on the fundamental problem of
seeking optimal cell average decomposition (OCAD), which arises from
constructing efficient high-order bound-preserving (BP) numerical methods
within Zhang--Shu framework. Since proposed in 2010, Zhang--Shu framework has
attracted extensive attention and been applied to developing many high-order BP
discontinuous Galerkin and finite volume schemes for various hyperbolic
equations. An essential ingredient in the framework is the decomposition of the
cell averages of the numerical solution into a convex combination of the
solution values at certain quadrature points. The classic CAD originally
proposed by Zhang and Shu has been widely used in the past decade. However, the
feasible CADs are not unique, and different CAD would affect the theoretical BP
CFL condition and thus the computational costs. Zhang and Shu only checked, for
the 1D $\mathbb P^2$ and $\mathbb P^3$ spaces, that their classic CAD based on
the Gauss--Lobatto quadrature is optimal in the sense of achieving the mildest
BP CFL conditions.
  In this paper, we establish the general theory for studying the OCAD problem
on Cartesian meshes in 1D and 2D. We rigorously prove that the classic CAD is
optimal for general 1D $\mathbb P^k$ spaces and general 2D $\mathbb Q^k$ spaces
of arbitrary $k$. For the widely used 2D $\mathbb P^k$ spaces, the classic CAD
is not optimal, and we establish the general approach to find out the genuine
OCAD and propose a more practical quasi-optimal CAD, both of which provide much
milder BP CFL conditions than the classic CAD. As a result, our OCAD and
quasi-optimal CAD notably improve the efficiency of high-order BP schemes for a
large class of hyperbolic or convection-dominated equations, at little cost of
only a slight and local modification to the implementation code.
