Crossposted on MO: https://mathoverflow.net/questions/406610/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli
Let $M$ be a two-dimensional Hadamard manifold. Consider three points $x_1, x_2, x_3 \in M$ and associate a function value $f_1, f_2, f_3 \in \mathbb{R}$ at each point.
We say a function $f \colon M \rightarrow \mathbb{R}$ interpolates the data $D=\{(x_i, f_i)\}_{i=1,2,3}$ if $f(x_i) = f_i$ for $i=1,2,3$.
Define the geodesically convex function $F_D \colon M \rightarrow \mathbb{R}\cup \{+\infty\}$ by $$F_{D}(x) = \sup_F F(x) \quad \forall x \in M$$ where the $\sup$ is taken over all geodesically convex functions $F$ interpolating the data $D$. When $M$ is the Euclidean plane, $F_D$ is simply a linear function inside the convex hull of $x_1, x_2, x_3$, and is $+\infty$ elsewhere.
What is an explicit formula for $F_D$ when $M$ is the hyperbolic plane? More generally, have these functions been studied, and if so where can I find a reference?
My attempts/thoughts:
(1) It is easy to see that $F_D$ is $+\infty$ outside of the (geodesic) convex hull of $x_1, x_2, x_3$.
(2) Upper bound approach: We can use convexity of $F_D$ to give an upper bound on $F_D$ along the geodesics connecting $x_i$ and $x_j$, for each $i, j \in \{1,2,3\}$. Given a point $x$ in the interior of the (geodesic) convex hull of $x_1, x_2, x_3$, we can then determine an upper bound on $F_D(x)$ by considering a geodesic passing through $x$ and one of the $x_i$'s. One might conjecture that $F_D(x)$ equals the minimum of all such upper bounds. However, after doing this computation on a simple example, one can show that this is not the case (because the resulting function is not convex while $F_D$ is convex).
(3) Simpler set ups: Consider the symmetric set up where $f_1 = 1, f_2=f_3=0$ and $x_2, x_3$ are the base vertices of an isosceles triangle. Even in this case, it is not clear to me what $F_D$ is.
(4) Epigraph approach: We can probably determine $F_D$ from the (geodesic) convex hull of the set $\bigcup_{i=1,2,3} (f_i, x_i) \subset \mathbb{R}\times M$. However, it is not clear to me how to determine this convex hull (because in general it will not be a totally geodesic submanifold of Riemannian product $\mathbb{R} \times M$).
(5) Special convex functions: Perhaps $F_D$ is harmonic or some linear combination of Busemann functions or distance functions to a convex set. However, I have not been able to prove this.