Fixed Point Theorem For Retracts Of A Simplex

If $K$ is a retract of a simplex $\Delta$, then every continuous map $$ f:K\to K $$ has a fixed point.

References

• [MFoGT, Lem. 4.11.3] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. A retract of a simplex has the fixed point property.