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We are studying $d$-dimensional geometric problems that have algorithms with$1-1/d$ appearing in the exponent of the running time, for example, in the formof $2^{n^{1-1/d}}$ or $n^{k^{1-1/d}}$. This means that these algorithms performsomewhat better in low dimensions, but the running time is almost the same rall large values $d$ of the dimension. Our main result is showing that for someof these problems the dependence on $1-1/d$ is best possible under a standardcomplexity assumption. We show that, assuming the Exponential Time Hypothesis,  --- $d$-dimensional Euclidean TSP on $n$ points cannot be solved in time$2^{O(n^{1-1/d-\epsilon})}$ for any $\epsilon>0$, and  --- the problem of finding a set of $k$ pairwise nonintersecting$d$-dimensional unit balls/axis parallel unit cubes cannot be solved in time$f(k)n^{o(k^{1-1/d})}$ for any computable function $f$.  These lower bounds essentially match the known algorithms for these problems.To obtain these results, we first prove lower bounds on the complexity ofConstraint Satisfaction Problems (CSPs) whose constraint graphs are$d$-dimensional grids. We state the complexity results on CSPs in a way to makethem convenient starting points for problem-specific reductions to particular$d$-dimensional geometric problems and to be reusable in the future for furtherresults of similar flavor.

The limited blessing of low dimensionality: when $1-1/d$ is the best possible exponent for $d$-dimensional geometric problems