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Every $n$-point metric space embeds into ${\ell }_1^{O(\log n)}$ with distortion $O(\log n)$. Apply the Bourgain embedding theorem, reduce its Euclidean dimension to $O(\log n)$ with the Johnson–Lindenstrauss lemma, and use an Almost-isometric Gaussian embedding from l2 into l1.

The Bourgain embedding theorem states that every $n$-point metric space embeds into a Hilbert space with distortion at most $C\log n$; the random-subset construction may be taken to have dimension $O((\log n{)}^2)$.

We use the following finite Frechet embedding lemma. For every integer $q\ge 2$, an $n$-point metric space $X$ has subsets $A_1,\dots ,A_k$ such thatand, for every $x,y\in X$, some $A_i$ satisfiesHere is the probabilistic proof. Put $\vartheta =n^{-1/q}$. For every level $i=1,\dots ,q$, independently formrandom subsets $A_{i,j}$ by including each point with probability $p_i=\min \{1/2,n^{-i/q}\}$.

Fix $x,y$ and put $\Delta =d(x,y)/(2q-1)$. Form $q+1$ open balls $B_0,\dots ,B_q$ by taking $B_t$ to have radius $t\Delta$ and center $x$ for even $t$ and $y$ for odd $t$. Consecutive balls are disjoint because their radii sum to at most $(2q-1)\Delta =d(x,y)$. Partition the cardinality interval $[1,n]$ intoAmong the $q+1$ cardinalities $|B_t|$, either two consecutive ones lie in one interval, or some consecutive pair decreases. In either case there are $i,t$ such that, after orienting the pair appropriately,At level $i$, a random set hits $B_t$ with probability at least $\vartheta /3$ and misses the disjoint $B_{t+1}$ with probability at least $1/4$. These events are independent, and when both occur the two distance-to-set values differ by at least $\Delta$. Thus one sample succeeds with probability at least $\vartheta /12$. All $m$ samples at that level fail with probability at mostA union bound over the fewer than $n^2/2$ unordered pairs leaves a simultaneous successful choice. There are $qm\le Cq{n}^{1/q}\log n$ coordinates, proving the lemma.

DefineEvery distance-to-set coordinate is one-Lipschitz, while the lemma supplies the lower bound. Henceproving the Low-dimensional Frechet embedding into linfinity.

Take $q=\lceil \log n\rceil$. Then $n^{1/q}\le e$, $k=O((\log n{)}^2)$, and the distortion into ${\ell }_{\infty }^k$ is $O(\log n)$. Sincethe identity ${\ell }_{\infty }^k\to {\ell }_2^k$ has distortion $O(\log n)$. Composition gives

Finally let a $d$-regular expander $G$ embed into ${\ell }_{\infty }^k$ with distortion at most $\alpha$. For $1<p<\infty$, the identityhas distortion at most $k^{1/p}$. Therefore the assumed lower bound givesFor $k\ge e^2$, choose $p=\log k$, so $k^{1/p}=e$ andAdjusting the constant handles bounded $k$, and exponentiation proves the Dimension lower bound for an expander embedded in linfinitywhere $c>0$ depends only on $d$ and $h$.

The Johnson–Lindenstrauss lemma says that for $0<\epsilon <1$, every set of $N$ points in a Hilbert space admits a linear map into ${\ell }_2^m$, where $m\le C{\epsilon }^{-2}\log N$, such that every pairwise distance is multiplied by a factor in $[1-\epsilon ,1+\epsilon ]$.

Let $S$ be a maximal $\delta$-separated subset of the unit sphere $S_E$. Maximality makes it a $\delta$-net. The translates $s+(\delta /2)B_E$, $s\in S$, have disjoint interiors and lie in $(1+\delta /2)B_E$. Comparing $n$-dimensional volumes givesso

Put $M=\Vert T\Vert$. For every $x\in S_E$, choose $s\in S$ with $\Vert x-s\Vert \le \delta$. Thenand taking the supremum givesThe reverse triangle inequality givesThus $T$ is injective and the net estimate for a linear operator yields

For a standard normal $Y$, completing the square givesWith $\beta =\sqrt{2/\pi }$, setNow $H(0)=H^{\prime }(0)=0$. Its second derivative is bounded above on $[0,1]$, so Taylor's theorem gives $H(u)\le C{u}^2$ there. For $u\ge 1$, the elementary bound $\Phi (u)\le 1$ givesThis proves the first subgaussian concentration of the absolute Gaussian average estimate. The second estimate is supplied in the question.

For fixed $x\ne 0$, rotational invariance of a Gaussian vector giveswhere the $Y_i$ are independent standard normals. HenceFor the upper tail, the Chernoff bound and the preceding moment estimate giveChoosing $u=\beta \delta /(2C)$ gives $e^{-c{\delta }^{2}k}$; the supplied negative-moment bound gives the same lower-tail estimate. Therefore

Fix $\epsilon >0$ and choose $0<\delta <1/3$ so thatTake a $\delta$-net of $S^{n-1}$ with at most $(3/\delta {)}^n$ points. If $k\ge C_{\epsilon }n$, the union bound and the concentration estimate show that with positive probability the random map satisfies the required inequalities simultaneously on the net. The net estimate then proves the Almost-isometric Gaussian embedding from l2 into l1 with distortion below $1+\epsilon$.

For the final claim, first apply the Bourgain embedding theorem to the given $n$-point metric space, obtaining distortion $O(\log n)$ in Euclidean dimension $O((\log n{)}^2)$. Apply the Johnson–Lindenstrauss lemma to its $n$ image points, reducing the dimension to $m=O(\log n)$ at constant additional distortion. Finally apply the preceding Gaussian construction with $k=O(m)=O(\log n)$. The composition proves the Low-dimensional L1 embedding of a finite metric space: