Longest convex chain ($L_n$)

For $n$ points in a triangle and two fixed endpoint vertices, the longest convex chain length is the largest number of sampled points in a convex chain joining those endpoints. It is a certifiable function: a chain of length $a$ supplies a certificate of exactly $a$ point coordinates. Changing one point changes the maximum by at most one, since deleting that point from any optimal chain loses at most one vertex. For independent uniform points, its median is $O(n^{1/3})$ and its fluctuation scale about the median is at most $n^{1/6}$, as proved by convex-chain probability in a triangle and convex-chain median concentration.

Let $m$ be an integer median for the longest convex chain length of independent uniform points in a triangle. If $L_n(x)\ge a$, fix a certificate of $a$ coordinates. Every configuration with $L_n\le b$ must disagree with at least $a-b$ of those coordinates; equal weights $1/\sqrt{a}$ prove Talagrand convex distance separation at least $(a-b)/\sqrt{a}$. Talagrand's convex distance inequality then gives $\mathbb{P}(L_n\ge a)\mathbb{P}(L_n\le b)\le e^{-(a-b{)}^2/(4a)}$. Set $b=m$, then $a=m$, in the two respective applications to obtain the displayed tails for positive integer $t$. The exact chain-counting bound gives $m\le C{n}^{1/3}$. Hence any interval centred at $m$ with length $\omega (n)n^{1/6}$, where $\omega (n)\to \infty$, contains $L_n$ with high probability. No upper restriction on the growth of $\omega$ is required.