13.2 Supporting Hyperplanes
Definition 13.2.1 (Supporting Hyperplanes).label Let $C\suf\R^{n}$ be a convex set, $C\neq\emp,\R^{n}$. We call a hyperplane $H=\curl{x\in\R^n:\inner{x}{a}=\al}$ a supporting hyperplane if
Equivalently we can require $\cl{C}\cap H\neq\emp$. The closed half-space $H^{-}$ is called the supporting half-space.
Example 13.2.2.label Let $C\suf\R^{n}$ be the ball then $C$ is convex and $H\cap \cl{C}=\curl{p}$ is unique. In fact given sufficient regularity (the normal vector is always defined) on $\pd C$ we always have uniqueness.
Definition 13.2.3 (Convex Body).label A convex body in $\R^{n}$ is a compact, convex set $K\suf\R^{n}$ with nonempty interior.
Proposition 13.2.4.label Let $K\suf\R^{n}$ be a convex body then every boundary point admits a supporting hyperplane.
Proof. Let $x_{0}\in\pd K=\pd (K^{c})$. Let $\curl{y_i}_{i\in\N}\suf K^{c}$ such that $y_{i}\to x_{0}$. BY Theorem 13.1.25 each $y_{i}\nin K$ can be strongly separated from $K$. So for each $i\in\N,\exists a_{i}\in \mathbb{S}^{n-1},\al_{i}$ such that
By compactness of $\mathbb{S}^{n-1}$ there exists a converging subsequence $a_{i_j}\to a\mathbb{S}^{n-1}$. For each $x\in K$ passing to the limit gives
Since $x_{0}\in K$ we have $H\define \curl{x\in\R^n:\inner{x}{a}=\inner{x_0}{a}}$ is a supporting hyperplane of $K$ through $x_{0}$.$\square$
Theorem 13.2.5.label Let $K\suf\R^{n}$ be a convex body then
In this sense we say the family of supporting hyperplanes determines $K$ completely.
Proof. The inclusion $\suf$ is immediate. Suppose $y\nin K$ then by Theorem 13.1.25, after choosing orientation of the normal vector, $\exists a\in\R^{n}\del\curl{\vec{0}}$ such that
By compactness of $K$, the supremum is attained at some $x_{a}\in K$. Hence
is a supporting hyperplane of $K$ and $y\nin H^{-}_{a}=\curl{x\in\R^n:\inner{x}{a}\leq\MAX{z\in K}\inner{z}{a}}$ as desired.$\square$
Definition 13.2.6 (Support Function).label Let $C\suf\R^{n}$ be a closed convex set. The support function of $C$ is
If $C$ is compact then $h_{C}(y)=\MAX{x\in C}\inner{x}{y}$. By homogeneity it suffices to consider the restriction $h_{C}|_{\mathbb{S}^{n-1}}:\mathbb{S}^{n-1}\to\R$. Indeed we can extend $\cl{h_C}:\R^{n}\to\R$
For a convex body $K$ and $y\neq 0$ the hyperplane
is a supporting hyperplane of $K$ and
is its supporting half-space. If $\norm{y}{}=1$, then $h_{K}(y)$ is the signed distance from the origin to $H_{K}(y)$; when $h_{K}(y)\geq 0$ it is the usual distance. Hence we can write Theorem 13.2.5 as
Theorem 13.2.7 (Dual Representation for Closed Convex Sets).label Let $C\suf\R^{n}$ be a nonempty closed convex set then
where $h_{C}(y)=\infty$ imposes no restriction.
Proof. check notes$\square$
Corollary 13.2.8.label Let $C,D\suf\R^{n}$ be nonempty closed convex sets then $h_{C}\leq h_{D}\iff C\suf D$ (on $\R^{n}$ or $\mathbb{S}^{n-1}$). In particular $h_{C}=h_{D}\iff C=D$
Remark 13.2.9 (Application in Differential Geometry).label Given a strictly convex closed set $C$, we can describe the curvature of $\pd C$ in terms of the support function. (Schneider)
Post a Comment