  We present two geometric applications of heat flow methods on the discrete
hypercube $\{-1,1\}^n$. First, we prove that if $X$ is a finite-dimensional
normed space, then the bi-Lipschitz distortion required to embed $\{-1,1\}^n$
equipped with the Hamming metric into $X$ satisfies
$$\mathsf{c}_X\big(\{-1,1\}^n\big) \gtrsim \sup_{p\in[1,2]}
\frac{n}{\mathsf{T}_p(X) \min\{n,\mathrm{dim}(X)\}^{1/p}},$$ where
$\mathsf{T}_p(X)$ is the Rademacher type $p$ constant of $X$. This estimate
yields a mutual refinement of distortion lower bounds which follow from works
of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020) for
low-dimensional spaces $X$. The proof relies on an extension of an important
inequality of Pisier (1986) on the biased hypercube combined with an
application of the Borsuk-Ulam theorem from algebraic topology. Secondly, we
introduce a new metric invariant called metric stable type as a functional
inequality on the discrete hypercube and prove that it coincides with the
classical linear notion of stable type for normed spaces. We also show that
metric stable type yields bi-Lipschitz nonembeddability estimates for weighted
hypercubes.
