  The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing
where the input is a set of $d$-dimensional rectangles, and the goal is to pack
them into unit $d$-dimensional cubes efficiently. It is NP-Hard to obtain a
PTAS for the problem, even when $d=2$. For general $d$, the best-known
approximation algorithm has an approximation guarantee exponential in $d$,
while the best hardness of approximation is still a small constant
inapproximability from the case when $d=2$. In this paper, we show that the
problem cannot be approximated within $d^{1-\epsilon}$ factor unless NP=ZPP.
  Recently, $d$-dimensional Vector Bin Packing, a closely related problem to
the GBP, was shown to be hard to approximate within $\Omega(\log d)$ when $d$
is a fixed constant, using a notion of Packing Dimension of set families. In
this paper, we introduce a geometric analog of it, the Geometric Packing
Dimension of set families. While we fall short of obtaining similar
inapproximability results for the Geometric Bin Packing problem when $d$ is
fixed, we prove a couple of key properties of the Geometric Packing Dimension
which highlight fundamental differences between Geometric Bin Packing and
Vector Bin Packing.
