Abstract : It is shown that for every $p∈(2,∞)$ there exists a doubling subset of $L_p$ that does not admit a bi-Lipschitz embedding into $\R^k$ for any $k\in \N$.
Vincent Lafforgue, Assaf Naor. A doubling subset of $L_p$ for $p>2$ that is inherently infinite dimensional. Geometriae Dedicata, Springer Verlag, 2014, 172 (1), pp.387-398. ⟨10.1007/s10711-013-9924-4⟩. ⟨hal-01144783⟩