A $2$-dimensional real Banach space with constant of analyticity less than one
Jorge Tomás Rodríguez
TL;DR
This work shows that the radius of analyticity can be strictly smaller than the radius of uniform convergence for real analytic functions on a finite-dimensional Banach space. By leveraging Bochnak's complexification constants and a constructive use of complexification, the authors prove $\mathcal{A}(\ell_1^2(\mathbb{R})) < 1$, providing the first non-trivial upper bound for the constant of analyticity in this setting. The main technique involves building a power series with unit convergence radius at the origin but deliberately restricting analytic extendability near a shifted point, exploiting differences between real and complex analyticity. The result separates real-analytic behavior from complex-analytic intuition and offers a concrete quantitative bound on $\mathcal{A}(X)$ for a notable finite-dimensional space.
Abstract
We show that on the real $2$-dimensional Banach space $\ell_1^2$ there is an analytic function $f:B_{\ell_1^2}\rightarrow \mathbb{R}$ such that its power series at origin has radius of uniform convergence one, but for some $a\in B_{\ell_1^2}$ the power series centred at that point has radius of uniform convergence strictly less than $1-\|a\|$. This result highlights a fundamental distinction in real analytic functions (compared to complex analytic functions), where the radius of analyticity can differ from the radius of uniform convergence. Moreover, this example provides the first non-trivial upper bound for the constant of analyticity.