These are continuous piecewise affine functions. Remarkably, they all can be written in max min form with piecewise affine components. We have shapes of the form you are asking (pyramids) in

http://ideas.repec.org/a/cup/macdyn/v10y2006i01p77-99_05.html
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=361228

These are applications of known results to economics, but I'd like to note that the results are due to

V.V. Gorokhovik and O.I. Zorko, “Piecewise afﬁne functions and polyhedral sets,” Optimization, vol. 31, pp.
209–221, 1994.

and not to Ovchinakov as claimed by my paper.

Now for your particular pyramid:
a) Let $f_1,f_2,f_3: \mathbb{R^2}\to \mathbb{R}$ be three affine functions each of the form
$f_i(x,y)=\alpha_1 x \beta_2 y + \delta_i$. These define the surfaces of the pyramid.
b) Let $g \mathbb{R^2}\to \mathbb{R}$ be the function $g(x,y)=0$.

Let
$$
f(x,y) = (\min_{i} f_i(x,y))\max g(x,y) =(\min_{i} f_i(x,y))^+
$$
which is a an interesting call option.:-)