Isometries without linear shadows: failure of Figiel's theorem beyond local convexity
A classical theorem of Figiel asserts that every isometric embedding $φ\colon X\to Y$ between real Banach spaces, with $φ(0)=0$, admits a linear contractive left inverse on the closed linear span of $φ(X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $φ\colon X\to Y$ such that $φ(0)=0$ and the closed linear span of $φ(X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $φ(X)$ which is a left inverse of $φ$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.