Suppose $v_1,\ldots, v_k \in \mathbb{R}^n$ are vector with all coordinates non-negative. How to explicitly calculate: $$ \max_{x\geqslant 0, ||x||_1=1} \min_{1\leqslant i \leqslant k} <x,v_i>$$
Remark1: $x\geqslant 0$ means that all all coordinates of $x$ are non-negative and $||x||_1=1$ means that $\sum_{i=1}^{n}x_i=1$.
Remark2: In terms of game theory, my problem is to calculate the value of a game with the 'row' and 'column' player and a matrix with non-negative entries.
Your problem can be written more concisely in the form
$$\max_{x \in \mathcal X}\min_{y \in \mathcal Y}\langle y, Ax\rangle,$$ where $A = \begin{bmatrix}v_1\\v_2\\\ldots\\v_k\end{bmatrix} \in \mathbb R^{k \times n}$, $\mathcal X := \{x \in \mathbb R^n | x \ge 0, \|x\|_1 \le 1\}$ and $\mathcal Y := \Delta_k$, the $k$-simplex. Your problem is thus to compute Nash-equilibrium for a two-player matrix game with payoff matrix $A$, and strategy profiles $\mathcal X$ and $\mathcal Y$. This can be done using a linear-programming solver, or more advanced methods of Nesterov (excessive-gap technique, etc.), Chambolle-Pock, etc. In these latter cases, note that there are linear time algorithms due to Duchi, Condat, etc., for pojection onto either of the strategy profiles.
N.B.: $\min_{1 \le i \le k}z_k = \min_{y \in \Delta_m}\langle y, z \rangle$.