Smooth maximum

Summary

In mathematics, a smooth maximum of an indexed family x₁, ..., x_n of numbers is a smooth approximation to the maximum function $\max(x_1,\ldots,x_n),$ meaning a parametric family of functions $m_\alpha(x_1,\ldots,x_n)$ such that for every $\alpha$, the function $m_\alpha$ is smooth, and the family converges to the maximum function $m_\alpha \to \max$ as $\alpha\to\infty$. The concept of smooth minimum is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, $m_\alpha \to \max$ as $\alpha \to \infty$ and $m_\alpha \to \min$ as $\alpha \to -\infty$. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.