# About Proof of Theorem 21.3 in Mathematics for Economists by Simon and Blume

## Background Information

I am studying concave and convex functions using Mathematics for Economists by Simon and Blume. I have difficulties understanding their proof of the following theorem:

Theorem 21.3$$\quad$$ Let $$f$$ be a $$C^1$$ function on a convex subset $$U$$ of $$\mathbb{R}^{\mathbf{n}}$$. Then, $$f$$ is concave on $$U$$ if and only if for all $$\mathbf{x}$$, $$\mathbf{y}$$ in $$U$$: \begin{align*} f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x}); \end{align*} that is, \begin{align*} f(\mathbf{y}) - f(\mathbf{x}) \leq \frac{\partial f}{\partial x_1}(\mathbf{x})(y_1 - x_1) + \dots + \frac{\partial f}{\partial x_n}(\mathbf{x})(y_n - x_n). \end{align*}

Here is the proof in textbook:

Proof$$\quad$$ Let $$\mathbf{x}$$ and $$\mathbf{y}$$ be arbitrary points in $$U$$. Let \begin{align*} g_{\mathbf{x}, \mathbf{y}}(t) & \equiv f(t\mathbf{y} + (1 - t)\mathbf{x}) \\ & = f(x_1 + t(y_1 - x_1), \dots, x_n + t(y_n - x_n)). \end{align*} Then, by the Chain Rule, \begin{align*} g'_{\mathbf{x}, \mathbf{y}}(t) = \sum_{i = 1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{x} + t(\mathbf{y} - \mathbf{x}))(y_i - x_i) \end{align*} and \begin{align*} g'_{\mathbf{x}, \mathbf{y}}(0) = \sum_{i = 1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{x})(y_i - x_i) = Df(\mathbf{x})(\mathbf{y} - \mathbf{x}). \end{align*} By Theorem 21.1 and 21.2, $$f$$ is concave if and only if every such $$g_{\mathbf{x}, \mathbf{y}}$$ is concave if and only if for every $$\mathbf{x}$$, $$\mathbf{y} \in U$$, \begin{align*} g_{\mathbf{x}, \mathbf{y}}(1) - g_{\mathbf{x}, \mathbf{y}}(0) \leq g'_{\mathbf{x}, \mathbf{y}}(0)(1 - 0) = g'_{\mathbf{x}, \mathbf{y}}(0) \end{align*} if and only if for every $$\mathbf{x}$$, $$\mathbf{y} \in U$$, \begin{align*} f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x}). \end{align*}

The proof uses the following two theorems, Theorems 21.1 and 21.2:

Theorem 21.1$$\quad$$ Let $$f$$ be a function defined on a convex subset $$U$$ of $$\mathbb{R}^{\mathbf{n}}$$. Then, $$f$$ is concave if and only if its restriction to every line segment in $$U$$ is a concave function of one variable.

Theorem 21.2$$\quad$$ Let $$f$$ be a $$C^1$$ function on an interval $$I$$ in $$\mathbb{R}$$. Then, $$f$$ is concave on $$I$$ if and only if \begin{align*} f(y) - f(x) \leq f'(x)(y - x) \quad \forall x, y \in I. \end{align*}

## My Question

I have problem with the part "By Theorem 21.1 and 21.2, $$f$$ is concave if and only if $$g$$ is concave if and only if $$g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$$." Apparently, "$$f$$ is concave if and only if $$g$$ is concave" is by Theorem 21.1, because $$g$$ is a restriction of $$f$$ to any line segment in $$U$$.

But, why is it that "$$g$$ is concave if and only if $$g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$$"? I guess this is supposed to be based on Theorem 21.2, because $$g$$ is a $$C^1$$ function on an interval $$I = [0,1]$$ in $$\mathbb{R}$$. However, if this is the case, then shouldn't it have written: $$g$$ is concave (on $$[0,1]$$) if and only if \begin{align*} g(p) - g(q) \leq g'(q)(p - q)\quad \forall p, q \in [0,1]? \end{align*} Why is $$g(1) - g(0)$$ sufficient?

Did I miss anything or is this proof wrong? Could someone please help me? Thanks a lot in advance!

What you call "$$g(1)-g(0)$$" is sufficient because it is actually a condition on $$g_{\mathbf{x}, \mathbf{y}}(1) - g_{\mathbf{x}, \mathbf{y}}(0)$$, which is required to hold for all $$x$$ and $$y$$. In other words, your suggestion to require $$g(p) - g(q) \leq g'(q)(p - q)\,\,\forall p,q\in[0,1]$$ is already contained in the proof's requirement, since for given $$x$$ and $$y$$ any choice of $$p$$ and $$q$$ just results in another (shorter) line segment between some point $$a$$ and some point $$b$$ in $$U$$ which can also be represented by setting $$x=a$$ and $$y=b$$, scaling again by $$t\in[0,1]$$ between these two endpoints.