For univariate functions, one way to test for convexity is to take the second derivitive ($f$ is convex if $\partial^{2} f/\partial^{2} x \geq 0$). Alternatively, one could assess convexity using the formula below. If $1\geq t \geq 0$, then a function $f$ is convex iff:
$$f(tx_1 + (1-t)x_2)\leq tf(x_1)+(1-t)f(x_2)$$
Now suppose we have a multivariate function (for example, a function with two inputs $f(x,y)$). I want to know if the cross partial $\partial^{2} f/\partial x\partial y >0$ (also a second derivative). Is there a similar way of testing for a positive cross-partial using a term such as $t$ in the formula above?