Rigidity of algebraic sets with small distortion

We prove that for positive integers $k$, $n$, and $d>1$, there is a maximal $c(k,n,d)>1$ such that if $X\subset {\mathbb{C}}^n$ is a connected $k$-dimensional complex algebraic set with degree at most $d$ and $dist(X)<c(k,n,d)$, then $X$ must be a complex affine linear subspace, where $dist(X)$ denotes the distortion of $X$ and is given by the supremum of the ratios $d_{inner}(p,q)\mathrm{/}{d}_{outer}(p,q)$ for all $p,q\in X$ with $p≠q$. We also prove that $c(1,n,d)=\sqrt{2}$ for any positive integers $k$ and $n$. This result provides a sharp rigidity phenomenon for curves in the $c$-LNE category, showing that near-isometric behavior of the inner and outer metrics forces the curve to be as simple as possible.