For convex function $c:\mathbb{R^2}\rightarrow\mathbb{R}$, the boundaries of the sets $S_t=\{x:c(x)\le t,x\in \mathbb{R^2}\},~t\in\mathbb{R}$ are level contours such as are drawn on topographic maps. Such a map is shown below.
I don't really understand this statement or the map. Some basic explanation or intuition on it would be helpful.
Seems a bit wrong to me,
You understand that a convex function is one such that any straight line between two points on its graph goes entirely above the graph and intersects it only in those two points
Now to get level contours of the map one should instead consider the set $S_t = \{\,\, p = (x,\, y,\, z)\,\, st: c(x,\,y)=t\,\}$
Now is it clear to you that this set considers the countour subset of the functions graph along which $c$ has constant value: $t$, i.e it is like taking out a level curve which marks the functions graph reaching a certain height t exactly at/along it.