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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!

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1 Answer 1

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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.

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