Suppose that $(x,y,z)\in[0,1]^3$ for simplicity. It is easy for me to interpret "$x+y<z$": this just means "$z$ is larger than the sum of x and y." Obviously "$e^{x+y}<e^z$" has the same interpretation. I'm having trouble interpreting "$e^x+e^y<e^z.$" Any suggestions for an intuitive interpretation of this inequality would be greatly appreciated!
An observation: Plotting $\left\{(x,y,z)\in[0,1]^3:e^x+e^y<e^z\right\}$ in MATLAB seems to suggest that in order to be in this set, $z$ must be larger than $x+y$ plus some additional buffer. Intuition behind why this happens?
the whole thing with equality is a surface. Here is a picture of level curves of $e^x + e^y$
Note that the level curves are identical, take the one that goes through the origin and shift by vector $(t, t).$ In the picture use $(\frac{1}{2}, \frac{1}{2}).$
Here it is with some diagonal lines drawn. If you tilt your head $45^\circ$ so that the diagonal lines appear to be horizontal, the idea that the curves are identical becomes more believable. I do not know how to rotate a jpeg or png by $45^\circ$