Generalization of convexity into higher dimensions.

Let $f: \mathbb{R}^d \rightarrow \bar{\mathbb{R}}$ be a convex function. For any sequence of vectors $x_1 \dots x_n \in \mathbb{R}^d$ and any sequence of non-negative scalars $\alpha_1 \dots \alpha_n \geq 0$ such that $\sum_{i=1}^n \alpha_i = 1$ we have:

$$ f \left( \sum_{i=1}^n \alpha_i x_i\right) \leq \sum_{i=1}^n \alpha_i f(x_i) $$

Visualize the following:

Source: Mark Reid.

The right-side of the above inequality lies within the polygon created by connecting points $x_1 , x_2, x_3, x_4$. Every point within that polygon is lower bounded at its associated $\hat{x}$ by the function itself (the left-side of the inequality).

The example he uses ($\hat{x} = \mathbb{E}[x]$) shows this inequality holds.

<aside> 💡 Note: While the above example is using $x_i \in \mathbb{R}^1$, this extends to all higher-dimensional spaces for any sequence of $x$’s. This may be difficult to convince yourself of, but you can prove it via induction.

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