Control of local interior-to-boundary ratios

Prove that, for sufficiently fine contact-point decompositions, every maximizing configuration in the general compact-convex-set optimization has uniformly small local ratios m_i/s_i between interior points and boundary vertices in each subtriangle.

Background

The proposed extension from the bi-pointed triangle to a general convex set requires showing that the number of interior points assigned to each small triangle is negligible relative to the number of boundary vertices assigned there.

The authors argue heuristically that the triangles become small as the number of contact directions grows, making large local ratios unlikely, but they do not prove the required uniform statement. This unresolved estimate is needed before the expansion of β near zero can be used in the optimization.

References

We may then conjecture that for any $\epsilon>0$, if $d$ large enough, for $(s[d],m[d])$ maximizing $f_{c[d]}$, $\max m_i/s_i\leq \epsilon$.

— 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