https://en.wikipedia.org/wiki/Concave_function According to the wikipedia article, a function f is said to be concave for any x and y and for any $ \alpha \in [0,1]$, the following holds: $ f((1-\alpha)x+\alpha y) \geq (1-\alpha)f(x) + \alpha y$.
I can imagine in two dimensional space that a line between any two points of the function is below the function value. So, as shown in the wikipedia picture, the line has two cross points b/w the function and clearly when $\alpha= 0, 1$, the inequality becomes f(x) = f(x) or f(y) = f(y)
And the function is called strictly concave if
$ f((1-\alpha)x+\alpha y) > (1-\alpha)f(x) + \alpha y$ holds for any $\alpha \in (0, 1)$ and $x\neq y$.
What is the pictorial difference between non-strictly concave and strictly concave function? What happens in the picture in the wikipedia for strict case? I could not get the point from Visual difference between strictly concave and not strictly concave
A line is concave, but not strictly concave. Consequently, the graph of a concave function can include linear segments, but the graph of a strictly concave function cannot contain a linear segment. For instance, the following graph is of a concave function, but that function is not strictly concave.
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