Suppose we are given a smooth (i.e., gradients are Lipschitz) function $f(x)$ and we try to find its minima via a simple gradient descent approach. Smoothness tells us that at each point $x$, the function $f(x)$ is bounded above by a quadratic. Finding the point that minimizes this quadratic immediately leads to the gradient descent update rule. Thus, there is a clear geometric interpretation of what smoothness buys us. Convexity and strong-convexity tell us that the function $f(x)$ is globally bounded below by a linear, and quadratic function, respectively. Is there a clear geometric interpretation of why this helps us? In terms of simple algebraic manipulations, standard proofs of convergence inform us that smoothness+convexity imply $||x_k-x^*||$ is a decreasing function of $k$, where $k$ is the iteration number, and $x^*$ is an optimal point. Again, is there a geometric interpretation of why the functional lower bounds supplied by either convexity or strong-convexity lead to the above phenomenon? Why does a quadratic lower bound (strong convexity) lead to better convergence than a linear lower bound (convexity)?
I understand the algebra that explains the above points, but it would be great to get more intuition.