Isometric canonical maps into free convex Banach lattices

For every $1<p<\infty$, we construct a three-dimensional real Banach lattice $X$ which is not $p$-convex with constant one, although its canonical positive map into the free $p$-convex Banach lattice with constant one is isometric. Dimension three is minimal. In every dimension $n\geq3$, such examples can be arbitrarily close to $\ell_p^n$ under the coordinate lattice isomorphism. An elementary example answers the necessity question raised by Marcel de Jeu and Xingni Jiang in 2025 for general convexity types; the three-dimensional construction gives a negative answer even for ordinary constant-one $p$-convexity. We characterize isometry by an exact factorization of dual functionals through two positive coordinates. For a Banach lattice, the criterion says that every sublattice of $X^*$ generated by two disjoint positive elements is $p'$-concave with constant one. The counterexample separates this two-generator condition from full concavity. At $p=\infty$, in contrast, isometry forces the original norm to be an AM-space norm.

GitHub repository: https://github.com/eulogica/free-lattice-isometric-maps