CAL1_L13

Line segments and paths

Definition 1(Definition).

Let $\mathbf{c},\mathbf{d}\in {\mathbb{R}}^n$. The line segment joining $\mathbf{d}$ and $\mathbf{d}$ is defined to be the subset

$$\{\mathbf{c}+t(\mathbf{d}-\mathbf{c})|t\in [0,1]\}.$$

Definition 2(Definition).

A path in ${\mathbb{R}}^n$ is an $n$-tuple $(x_1,\dots ,x_n)$ of continuous functions $x_1,\dots ,x_n:[\alpha ,\beta ]\to \mathbb{R}$, where $\alpha ,\beta \in \mathbb{R}$ and $\alpha <\beta$. The path is said to be from $(x_1(\alpha ),\dots ,x_n(\alpha ))$ to $(x_1(\beta ),\dots ,x_n(\beta ))$.

Definition 3(Definition).

$D\subseteq {\mathbb{R}}^n$ is said to be

1. convex if the line segment joining any two points of $D$ lies in $D$,
2. path-connected if any two points of $D$ can be joined by a path that lies in $D$.

Definition 4(Definition).

Let $D\subseteq {\mathbb{R}}^n$ and $f:D\to \mathbb{R}$ be any function. Also, let $A$ be a convex subset of $D$.
$f$ is convex on $A$ if for all $\mathbf{c},\mathbf{d}\in A$ and $t\in (0,1)$, we have

$$f(\mathbf{c}+t(\mathbf{d}-\mathbf{c}))\le f(\mathbf{c})+t(f(\mathbf{d})-f(\mathbf{c})),$$

$f$ is concave on $A$ if for all $\mathbf{c},\mathbf{d}\in A$ and $t\in (0,1)$, we have

$$f(\mathbf{c}+t(\mathbf{d}-\mathbf{c}))\ge f(\mathbf{c})+t(f(\mathbf{d})-f(\mathbf{c})).$$

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Local extrema and saddle points

Definition 5(Definition).

Let $\Gamma$ be a path in ${\mathbb{R}}^2$ given by $\mathbf{x}(t)$, $t\in [\alpha ,\beta ]$.

1. $\Gamma$ is said to pass through a point ${\mathbf{x}}_0\in {\mathbb{R}}^2$ if there is $t_0\in (\alpha ,\beta )$ such that $\mathbf{x}(t_0)={\mathbf{x}}_0$.
2. If $x_1$ and $x_2$ are differentiable at $t_0\in (\alpha ,\beta )$ and ${\mathbf{x}}^{\prime }(t_0)\equiv (x_1^{\prime }(t_0),x_2^{\prime }(t_0))\ne 0$, we define the tangent of $\Gamma$ at $\mathbf{x}(t_0)$ to be ${\mathbf{x}}^{\prime }(t_0)$.
3. $\Gamma$ is called a regular path if the tangent is defined for all $t\in (\alpha ,\beta )$.

Definition 6(Definition).

Let $D\subseteq {\mathbb{R}}^2$ and let ${\mathbf{x}}_0\in D^{\circ }$. Let a path $\Gamma$ lie in $D$ and be given by $\mathbf{x}(t)$ for $t\in [\alpha ,\beta ]$, and let $\mathbf{x}(t_0)={\mathbf{x}}_0$ for some $t_0\in (\alpha ,\beta )$. Let $f:D\to \mathbb{R}$ be any function. Define $F:[\alpha ,\beta ]\to \mathbb{R}$ by $F(t)\equiv f(\mathbf{x}(t))$. We say that

1. $f$ has a local maximum at ${\mathbf{x}}_0$ along $\Gamma$ if $F$ has a local maximum at $t_0$,
2. $f$ has a local minimum at ${\mathbf{x}}_0$ along $\Gamma$ if $F$ has a local minimum at $t_0$.

Definition 7(Definition).

Suppose ${\Gamma }_1$ and ${\Gamma }_2$ are regular paths in ${\mathbb{R}}^2$ which pass through the same point ${\mathbf{x}}_0$. Then ${\Gamma }_1$ and ${\Gamma }_2$ are said to intersect transversally at ${\mathbf{x}}_0$ if their tangent vectors at ${\mathbf{x}}_0$ are defined and are not multiples of each other.

Definition 8(Definition).

Let $D\subseteq {\mathbb{R}}^2$ and let ${\mathbf{x}}_0$ be an interior point of $D$. We say that a function $f:D\to \mathbb{R}$ has

1. a local maximum at ${\mathbf{x}}_0$ if there is $\delta >0$ such that $B_{\delta }({\mathbf{x}}_0)\subseteq D$ and $f(\mathbf{x})\le f({\mathbf{x}}_0)$ for all $\mathbf{x}\in B_{\delta }({\mathbf{x}}_0)$,
2. a local minimum at ${\mathbf{x}}_0$ if there is $\delta >0$ such that $B_{\delta }({\mathbf{x}}_0)\subseteq D$ and $f(\mathbf{x})\ge f({\mathbf{x}}_0)$ for all $\mathbf{x}\in B_{\delta }({\mathbf{x}}_0)$,
3. a saddle point at ${\mathbf{x}}_0$ is there are regular paths ${\Gamma }_1$ and ${\Gamma }_2$ lying in $D$ and intersecting transversally at ${\mathbf{x}}_0$ such that $f$ has a local maximum along ${\Gamma }_1$ and a local minimum along ${\Gamma }_2$ at ${\mathbf{x}}_0$.