Characterization of interior points of a convex polytope

Let's assume wlog. $P$ has full rank by restricting ourselves to the affine hull if necessary. Let $x$ be a point in the interior of $P$.

Claim: For every $i$, there exists a convex combination $x = \sum_j \lambda^i_j x_j$ such that $\lambda^i_i \neq 0$.

Then $x = \sum_j \frac 1n(\sum_i\lambda^i_j)x_j$ is a positive convex combination of $x$, as desired.

Proof of claim: Since $x$ is in the interior of $P$, there exists some $\varepsilon > 0$ such that $y := x + \varepsilon(x - x_i)$ is in $P$. Hence there exists a convex combination $y = \sum_j\mu_jx_j$. Then $$x = \frac{1}{1+\varepsilon}(y + \varepsilon x_i) = \frac{\varepsilon}{1+\varepsilon}x_i + \sum_j\frac{1}{1+\varepsilon}\mu_j x_j,$$ so

$$\lambda_j^i := \frac{1}{1+\varepsilon}\begin{cases}\varepsilon+\mu_j&\text{ if }i=j\\\mu_j&\text{ else }\end{cases}$$ satisfies $x = \sum_j \lambda^i_j x_j$ and $\lambda^i_i \neq 0$.