On sharp isoperimetric inequalities on the hypercube

We prove the sharp isoperimetric inequality $$ \mathbb{E} \,h_{A}^{\log_{2}(3/2)} \geq \mu(A)^{*} (\log_{2}(1/\mu(A)^{*}))^{\log_{2}(3/2)} $$ for all sets $A \subseteq \{0,1\}^n$, where $\mu$ denotes the uniform probability measure, $\mu(A)^{*}=\min\{\mu(A), 1-\mu(A)\}$, $h_A$ is supported on $A$ and to each vertex $x$ assigns the number of neighbour vertices in the complement of $A$. The inequality becomes equality for any subcube. Moreover, we provide lower bounds on $\mathbb{E} h_{A}^{\beta}$ in terms of $\mu(A)$ for all $\beta \in [1/2,1]$, improving, and in some cases tightening, previously known results. In particular, we obtain the sharp inequality $\mathbb{E}h_{A}^{0.53}\geq 2 \mu(A)(1-\mu(A))$ for all sets with $\mu(A)\geq 1/2$, which allows us to refine a recent result of Kahn and Park on isoperimetric inequalities about partitioning the hypercube. Furthermore, we derive Talagrand's isoperimetric inequalities for functions with values in a Banach space having finite cotype: for all $f :\{-1,1\}^{n} \to X$, $\|f\|_{\infty}\leq 1$, and any $p \in [1,2]$ we have $$ \|Df\|_{p} \gtrsim \frac{1}{q^{3/2}C_{q}(X)} \|f\|_{2}^{2/p}\left(\log \frac{e\|f\|_{2}}{\|f\|_{1}}\right)^{1/q}, $$ where $\| Df\|_{p}^{p} = \mathbb{E} \| \sum_{1\leq j \leq n} x'_{j} D_{j} f(x)\|^{p}$, $x'$ is independent copy of $x$, and $C_{q}(X)$ is the cotype $q$ constant of $X$. Different proofs of the recently resolved Talagrand's conjecture will be presented.