# “Concave and Convex function”

## Definition

A real-valued function $f$ on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any $x$ and $y$ in the interval and for any $\alpha \in [0,1]$.

[f((1-\alpha) x+\alpha y) \geq(1-\alpha) f(x)+\alpha f(y)]

A function is called strictly concave if

[f((1-\alpha) x+\alpha y)>(1-\alpha) f(x)+\alpha f(y)]

for any $\alpha \in [0,1]$ and $x \neq y$

For a function $f: \mathbb{R} \rightarrow \mathbb{R}$, this second definition merely states that for every $z$ strictly between $x$ and $y$, the point $(z,f(z))$ on the graph of $f$ is above the straight line joining the points $(x,f(x))$ and $(y,f(y))$.

Convex is the opposite of concave case.

## YONG HUANG

Hi, I'm Yong Huang. I've recently graduated from Cornell Tech and obtained my master's degree, I shall start my Ph.D. in Computer Science this fall at UC Irvine. Thank you for visiting my site.