We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a \emph{Caratheodory set} and a \emph{Helly set} and certain decision versions of the \emph{\hsc\ problem} are \wone-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the \dSum\ problem. Our reductions also imply that the problems we consider cannot be solved in time~$n^{o(d)}$ (where $n$ is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. The technique of embedding \dSum\ into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial \wone-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.