Perturbation by a Euclidean regular polygon: comparison-disk bounds for degenerate comparison triangles

The auxiliary regular polygon. Put $c=2\epsilon \sin (\pi \mathrm{/}n)$ and $q=\cos (\pi \mathrm{/}n)\in (0,1)$. Euclidean chord midpoints show that the $k$-th iterate of the regular polygon has circumradius $\epsilon q^k$, edge length $c{q}^k$, length $nc{q}^k$ and energy $n{c}^2{q}^{2k}$. The bounded decreasing sequence $q^k$ has a limit by A monotone sequence converges if and only if it is bounded, and that limit equals $q$ times itself, hence is zero since $q<1$. Thus all four quantities tend to zero.

Product geometry and correct length bounds. The product geodesics and midpoint operation are componentwise by [F1]. For any product tuple $y=(y^X,y^E)$, its edge lengths are $\sqrt{a_i^2+b_i^2}$, where $a_i,b_i$ are the component edge lengths, so $E(y)=E(y^X)+E(y^E)$ and $L(y)\le L(y^X)+L(y^E)$. In general the lengths themselves do not satisfy an additive squared identity. The product has the same uniform CAT(1) radius $l$: a radius-$l$ product ball lies in the product of the CAT(1) $X$-ball and a convex Euclidean ball, which is CAT(1) by [F1]; the product ball is a radius-$l<\pi \mathrm{/}2$ ball in that CAT(1) chart, so is convex and CAT(1) by [F5]. Thus the finite construction of [F2], which uses the uniform radius but not ambient compactness, applies in this product.

The edge lower bound survives. Let $a={\min }_i{\xi }_i>0$. For $u\ge a$, $\sqrt{u^2+c^2}\mathrm{/}\sqrt{a^2+c^2}\le u\mathrm{/}a$, as follows by squaring. Summing gives $L(x^{\epsilon })\mathrm{/}{\min }_i{\xi }_i^{\epsilon }\le L(x)\mathrm{/}a\le 2n$. Thus every perturbed edge obeys ${\xi }_i^{\epsilon }\ge L(x^{\epsilon })\mathrm{/}(2n)$, exactly the hypothesis of the finite quantitative output; no strict slack in the original lower bound is required.

Strict inequalities for the actual face triples. The ear triples are an old regular-polygon vertex and the midpoints of its two incident edges; their Euclidean projections are noncollinear, since the two old incident edges are not collinear and their lengths are positive. The fan triples are three distinct vertices of the last regular polygon, also noncollinear. This holds at each finite iteration since $\epsilon q^k>0$. If $a,b,c$ are the three $X$-distances and $\alpha ,\beta ,\gamma$ their Euclidean counterparts, then $a+b\ge c$ and $\alpha +\beta >\gamma$; hence $\sqrt{a^2+{\alpha }^2}+\sqrt{b^2+{\beta }^2}\ge \sqrt{(a+b{)}^2+(\alpha +\beta {)}^2}>\sqrt{c^2+{\gamma }^2}$. Relabeling proves every strict triangle inequality for every ear and cap face.

Basin, mesh and finite cap. Componentwise iteration and step 1.2 give $L(f^k{x}^{\epsilon })\le L(f^{k}x)+nc{q}^k\to 0$. For sufficiently small $\epsilon$, the product mesh ${\max }_i\sqrt{{\xi }_i^2+c^2}$ is below $l$ and its length is below $2\pi$; choose a product mesh bound $h_{\epsilon }<l$ containing this tuple. The basin condition supplies an integer $m\ge 1$ with product iterate length $<2l$, and step 2.1 makes every face of that finite disk nondegenerate. The finite supplier's conclusions therefore apply without any compactness assumption on $X\times {\mathbb{R}}^2$.

The perturbed deficit. Write $r_{\epsilon }=L(x^{\epsilon })\mathrm{/}2$, ${\eta }_{\epsilon }=(\pi -r_{\epsilon })\mathrm{/}2$, and ${\mu }_{\epsilon }=\min \{r_{\epsilon }\mathrm{/}(2n),{\eta }_{\epsilon }\}$. By steps 3.1 and 1.3 and F2, there is an index $k=k(\epsilon )$ such that ${\zeta }_k^{\epsilon }\le ({\xi }_k^{\epsilon }+{\xi }_{k+1}^{\epsilon })\mathrm{/}2-\delta ({\eta }_{\epsilon },{\mu }_{\epsilon })$. The constants are universal functions of the perturbed length and $n$, not of the space or mesh. AC is used solely through the finite CAT(1) disk supplier.

Limit and conclusion. Take ${\epsilon }_j\downarrow 0$. The product edge lengths ${\xi }_i^{{\epsilon }_j}=\sqrt{{\xi }_i^2+c_j^2}$ converge to ${\xi }_i$, and the midpoint lengths converge to ${\zeta }_i$ because the midpoint operation is componentwise and the Euclidean midpoint-edge length is $c_{j}q$. Consequently all fixed-iterate lengths and energies converge to their original values, ${\eta }_{{\epsilon }_j}\to \eta$, ${\mu }_{{\epsilon }_j}\to \mu$, and $\delta ({\eta }_{{\epsilon }_j},{\mu }_{{\epsilon }_j})\to \delta (\eta ,\mu )$ by the finite supplier's continuity. Some index $k$ occurs infinitely often, since there are only finitely many indices; on that subsequence step 4.1 gives ${\zeta }_k\le ({\xi }_k+{\xi }_{k+1})\mathrm{/}2-\delta (\eta ,\mu )$. This proves the quantitative extension, including degenerate face triples; the perturbation energies vanish and the product comparison is the CAT(1) comparison of [F1].