Sobolev homeomorphisms and Brennan's conjecture

Let $Ω\subset \mathbb{R}^n$ be a domain that supports the $p$-Poincaré inequality. Given a homeomorphism $φ\in L^1_p(Ω)$, for $p>n$ we show the domain $φ(Ω)$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain $Ω' \subset\mathbb C$ with nonempty boundary and for any conformal homeomorphism $φ$ from the unit disc $\mathbb{D}$ onto $Ω'$ the complex derivative $φ'$ is integrable in the degree $s$, $-2<s<2/3$. If $Ω'$ is bounded than $-2<s\leq 2$. We prove that integrability in the degree $s> 2$ is not possible for domains $Ω'$ with infinite geodesic diameter.

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