A $2$-dimensional real Banach space with constant of analyticity less than one

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.