Extremal convex function conjecture for normalized Donaldson–Futaki invariant

Let $P$ be a polytope, let $u_T^{\mathrm{can}}$ be the canonical family, and define the convex function

$$q_T:=T\log u_T^{\mathrm{can}}.$$

Let $\mathrm{DF}(q)$ denote the Donaldson–Futaki invariant, let $\bar q=\int_Pq\,d\mu/\int_Pd\mu$, and let $\|\hat q\|_{L^2}=(\int_P(q-\bar q)^2d\mu)^{1/2}$. Extremal destabilizer conjecture. As $T\to\infty$, $q_T$ converges at least in the $L^2$-topology to a lower-semicontinuous convex function $q_{\mathrm{ext}}\in L^2(P)$ that minimizes

$$\frac{\mathrm{DF}(q)}{\|\hat q\|_{L^2}}=\frac{2\pi\int_{\partial P}q\,d\sigma+\bar s\int_Pq\,d\mu}{\left(\int_P(q-\bar q)^2d\mu\right)^{1/2}}$$

among all $L^2$-integrable convex functions on $P$. This proposes that the optimal destabilizer for the normalized Donaldson–Futaki invariant appears in the rescaled limit; the stated regularity is stronger than the known $L^2$-regularity of $q_{\mathrm{ext}}$.