Uniform approximation for local triangular probabilities

Derive a uniform exponential-order approximation for the triangular probabilities $\bQt_{s_i,m_i}$ in the multi-triangle decomposition, sufficiently uniform over the local ratios $m_i/s_i$, to justify replacing each probability by the asymptotic expression involving $\beta_{m_i/s_i}$ and thereby complete the general-convex-set conjectures.

Background

The general-case analysis decomposes configurations according to contact points and assigns each subtriangle s_i boundary points and m_i interior points. The desired optimization requires replacing the exact factors \bQt_{s_i,m_i} by their exponential asymptotics.

Although the paper notes that an asymptotic formula follows when x(n)/n tends to λ, the authors explicitly state that they have not established the uniform control needed when the ratios m_i/s_i vary across subtriangles, particularly because these ratios may be unbounded.

References

It is not difficult however to prove that if $x(n)/n\to \lambda\in [0,+\infty)$ then \bQt_{n,x(n)}=\bQt_{n,\floor{n\lambda}\exp( n o(1))=\exp(-2n\log(n)+n\beta_\lambda+no(1)) by adapting the proofs we presented for the asymptotics of $\bQt_{n,\floor{n\lambda}$, however, we have not found a complete argument to prove that the optimization of

— Conditioning random points by the number of vertices of their convex hull: the bi-pointed case  (2510.26330 - Marckert et al., 30 Oct 2025) in Section 'Construction of the conjectures in the case $\QK_{n,\floor{n\lambda}}$', Section 1