What exactly is meant by convex surfaces?

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Google search didn't show up. It just shows up information related to spherical mirrors everywhere.

Is there a way to intuitively (and maybe formally) define convex (and concave) surfaces around a point $P$?

The following is an extract from my book:

"Now, the surface $S$ being convex, the quantity $\cos (\hat{r},\hat{n})$ is positive for every element $d\omega$ and can not become null in any of them."

Here the author is referring to the solid angle formula at point $P$ due to a surface $S$:

$$dS=\dfrac{r^2}{\cos (\hat{r},\hat{n})}\ d\omega$$

where the symbols have their their usual meanings.