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Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ which is convex and satisfies $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0$. The last condition is equiva …

Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots, x_n+y_n) \leq f(x …

Continuous version of this Superhomogeneity of subadditive functions Let $f$ be a continuous function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f …

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying $f \geq 0$, $f(0,0) = 0$, $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \ …

Let $f\colon \mathbf{R} \times \mathbf{R}^+ \rightarrow \mathbf{R}$ be defined by $f(x,y) = \frac{x^2}{y}$. Let $X = \left\lbrace x_1, \dots, x_n\right\rbrace \subseteq \mathbf{R}$, $Y = \left\lbrace …