Let $G$ be an undirected graph.

- The eccentricity of a vertex $v$ of $G$, is the maximum distance between $v$ and any other vertex of $G$:$\;\;$ $\mathit{ecc}(v) = \max_{u}\mathit{dist}(v,u)$.
- The radius of $G$ is the minimum eccentricity of a vertex of $G$:$\;\;$$R(G) = \min_v \mathit{ecc}(v)$.
- The diameter of $G$ is the maximum eccentricity of a vertex of $G$: $\;\;$ $D(G) = \max_v \mathit{ecc}(v)$.

**Question:** Clearly we have $R(G)\leq D(G)$. Is it possible to improve this upper-bound of $R(G)$ in function of $D(G)$ on maximal planar graphs? For instance, if $G$ is *outerplanar*, then it is known that $R(G)\leq \lfloor\frac{D(G)}{2}\rfloor + 1$. Is the same relation valid for maximal planar graphs?

Obs1: An outerplanar graph is a planar graph which can be drawn in the plane in such a way that no two edges cross and all vertices belong to the outer-face of the drawing.

Obs2: A maximal planar (outerplanar) graph is a planar (outerplanar) graph whose addition of one edge destroys the property of being planar (outerplanar).