Two-dimensional real symmetric spaces with maximal projection constant

Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that ${\displaystyle \lambda \left(V\right)\le \lambda \left(V_n\right)}$ where ${\displaystyle V_n}$ is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that ${\displaystyle \frac{4}{\pi }=\lambda \left(l{₂}^{\left(2\right)}\right)\ge \lambda \left(V\right)}$ for any two-dimensional real symmetric space V.