In the book "Generalized Convexity and Optimization" by A.Cambini and L.Martein, they talked about the differentiable quasiconvex functions. For quasiconvex function that is differentiable and defined on a convex set, it can be categorized as:
Lemma: $f$ is quasiconvex on $S$ if and only if for all $\mathbf{x}_{1},\mathbf{x}_{2}\in S$, we have $$f(\mathbf{x}_{1})\geq f(\mathbf{x}_{2})\implies [\nabla f(\mathbf{x}_{1})]^{\intercal}(\mathbf{x}_{2}-\mathbf{x}_{1})\leq 0.$$
In page $41$, they say the followings:
A differentiable function is convex if and only if its graph lies on or above the tangent at any point of the graph. Similarly, a differentiable function is quasiconvex if and only if any of its level sets lies on or below the tangent at any point of the level set.
I understand what they mean for convex function. But, what does they mean here for the quasiconvex function? Do they mean the following?
Fixing $c\in\mathbb{R}$ and consider the level set $L_{c}(f)$. Let $\mathbf{x}_{0}\in L_{c}(f)$, define the tangent at $\mathbf{x}_{0}$ as $$P_{\mathbf{x}_{0}}(\mathbf{x}):=f(\mathbf{x}_{0})+[\nabla f(\mathbf{x_{0}})]^{\intercal}(\mathbf{x-x_{0}})=c+[\nabla f(\mathbf{x_{0}})]^{\intercal}(\mathbf{x-x_{0}}).$$ Then, $f$ is quasiconvex if and only if $c\leq P_{\mathbf{x}_{0}}(\mathbf{x})$ for all $\mathbf{x}\in S$ (and for all $c=c(\mathbf{x}_{0})$ and all $\mathbf{x}_{0}$).
Does this even make sense? For example, if $n=1$, then the level set is just a bunch of points lying on the real line. Then the tangent line at any such point on the curve $f(x)$ is above it as long as $f(x)>0$ at that point. This is clearly a misunderstanding..
Also, how could we prove this using the Lemma? For example, let us try to prove $(\Rightarrow)$ using the Lemma above. Suppose that $f$ is quasiconvex on a nonempty convex set $S$. This is equivalently to showing that $$[\nabla f(\mathbf{x_{0}})]^{\intercal}(\mathbf{x-x_{0}})\geq 0.$$ But the Lemma only provides $\leq 0$ even if I manage to show that $c=f(\mathbf{x}_{0})\geq f(\mathbf{x})$, which is clearly not true in general...
What is the geometric interpretation of this Lemma anyway? Thank you!!
The geometric interpretation is that global descent directions are local descent directions.
By global descent direction, I mean a direction which, if you travel in that direction, at some point you will be lower than where you started (even if at first you are going up). By local descent direction, I mean that if you start going in that direction a very short distance, you will immediately be lower than where you started.
Imagine that you are standing on a landscape, and there is another far away point on the landscape which is lower, which you want to get to. If you start walking straight towards the lower point, your first step will be going downhill.
(edited to incorporate comment in answer)