Let $(X_j,d_j)_{j \in \mathbb{N}}$ be a sequence of compact geodesic metric spaces converging in Gromov-Hausdorff distance to a compact [metric space](/page/Metric%20Space) $(X,d)$. Fix an integer $k \geq 2$ and a [closed set](/page/Closed%20Set) $C \subset \mathbb{R}^{k(k-1)/2}$. For a compact metric space $(Y,\rho)$, say that $(Y,\rho)$ satisfies the closed finite distance comparison condition determined by $C$ if, for every ordered $k$-tuple $(y_1,\dots,y_k) \in Y^k$, the distance vector

\begin{align*} D_Y(y_1,\dots,y_k) := \bigl(\rho(y_a,y_b)\bigr)_{1 \leq a < b \leq k} \in \mathbb{R}^{k(k-1)/2} \end{align*}

belongs to $C$.

If every $(X_j,d_j)$ satisfies this condition, then $(X,d)$ is a compact geodesic metric space satisfying the same condition. In particular, any triangle comparison property whose hypotheses and conclusion can be encoded by finitely many distances and a closed subset of a finite-dimensional Euclidean space is preserved under compact Gromov-Hausdorff limits.