A function $f: C \mapsto R \ $ for a convex set $C \subset R^{d} $ is said to be convex if for any $\lambda \in [0, 1]$ and any $x, y \in C$, the following holds
$$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1 - \lambda) f(y)$$
This definition for convexity is visualized geometrically as any line joining 2 points on a curve being above the curve for all values between $x$ and $y$.
Is there an equivalent notion of convexity for vector-valued functions $f: C \mapsto R^{m} \ $ for a convex set $C \subset R^{d} $?
What would be the geometric interpretation of this notion if any?
As suggested by @dohmatob, we can consider the convex function inequality component-wise.
That is the function $f: C \mapsto R^m$, where $C \subset R^d$ is geiven by,
$$f:x \mapsto (f_1(x), \ldots ,f_m(x))$$
With each $f_i(x)$ convex in the traditional sense.
A geometrical interpretation of this definition is not obvious to me.