Complex geodesics in convex domains and $\mathbb{C}$-convexity of semitube domains

In the paper the complex geodesics of a convex domain in ${\mathbb{C}}^n$ are studied. One of the main results of the paper provides certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ${\mathbb{C}}^n$. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The $\mathbb{C}$-convexity of semitube domains is also discussed.