On the real $X$-ranks of points of ${\mathbb{P}}^n(\mathbb{R})$ with respect to a real variety $X\subset {\mathbb{P}}^n$

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $\mathbb{R}$. For any $P\in {\mathbb{P}}^n(\mathbb{R})$ the real $X$-rank $r_{X,\mathbb{R}}(P)$ is the minimal cardinality of $S\subset X(\mathbb{R})$ such that $P\in ⟨S⟩$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.