Inapproximability for Metric Embeddings into R^d

We consider the problem of computing the smallestpossible distortion for embedding ofa given $n$-point metric space into $\R^d$, where $d$is \emph{fixed} (and small). For $d=1$, it was known thatapproximating the minimum distortion with a factor better than roughly $n^{1/12}$ is NP-hard.  From this resultwe derive inapproximability withfactor roughly $n^{1/(22d-10)}$ for every fixed $d\ge 2$,by a conceptually very simple reduction. However,the proof of correctness involves a nontrivialresult in geometric topology (whose current proof isbased on ideas due to Jussi V\"ais\"al\"a).For $d\ge 3$, we obtain a stronger inapproximabilityresult by a different reduction: assuming P$\ne$NP,no polynomial-time algorithm can distinguish betweenspaces embeddable in $\R^d$ with constant distortionfrom spaces requiring distortion at least $n^{c/d}$,for a constant $c≫0$. The exponent $c/d$has the correct order of magnitude, sinceevery $n$-point metric space canbe embedded in $\R^d$ with distortion$O(n^{2/d}\log^{3/2}n)$ and such an embeddingcan be constructed in polynomial time byrandom projection.For $d=2$, we give an example of a metric space thatrequires a large distortion for embedding in $\R^2$,while all not too large subspaces of it embed almost isometrically.