  In the Knapsack problem, one is given the task of packing a knapsack of a
given size with items in order to gain a packing with a high profit value. In
recent years, a connection to the $(\max,+)$-convolution problem has been
established, where knapsack solutions can be combined by building the
convolution of two sequences. This observation has been used to give
conditional lower bounds but also parameterized algorithms.
  In this paper we want to carry these results into higher dimension. We
consider Knapsack where items are characterized by multiple properties - given
through a vector - and a knapsack that has a capacity vector. The packing must
now not exceed any of the given capacity constraints. In order to show a
similar sub-quadratic lower bound we introduce a multi-dimensional version of
convolution as well. Instead of combining sequences, we will generalize this
problem and combine higher dimensional matrices. We will establish a few
variants of these problems and prove that they are all equivalent in terms of
algorithms that allow for a running time sub-quadratic in the number of entries
of the matrix.
  We further develop a parameterized algorithm to solve higher dimensional
Knapsack. The techniques we apply are inspired by an algorithm introduced by
Axiotis and Tzamos. In general, we manage not only to extend their result to
higher dimension. We will show that even for higher dimensional Knapsack, we
can reduce the problem to convolution on one-dimensional sequences, leading to
an $\mathcal{O}(d(n + D \cdot \max\{\Pi_{i=1}^d{t_i}, t_{\max}\log t_{\max}\}
))$ algorithm, where $D$ is the number of different weight vectors, $t$ the
capacity vector and $d$ is the dimension of the problem. Finally we also modify
this algorithm to handle items with negative weights to cross the bridge from
solving not only Knapsack but also Integer Linear Programs (ILPs) in general.
