What is the support function of an ellipse?

Question

I defined the support function $h_A:R^n→R$ of a non-empty closed convex set $A\subseteq \mathbb{R}^n$ as $$h_A(x)= \sup\left\{x⋅a |\ a \in A\right\}$$

Everything I know about this topic I found it. I have to calculate the support function of an ellipse $$\text{E=$\left\{(x,y) \in \mathbb{R}^2 \quad|\quad \frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1 \right\}$}$$

For $x=a\cos\theta ,\ y=b\sin\theta, \ 0\le\theta\lt2\pi\ $, how can calculate $h_E(\theta)$?

The result should be $h_E(\theta)=\left(a^2\cos^2\theta+b^2\sin^2\theta \right)^\frac12$ but my calculations lead me to have $h_E(\theta)=\sup\{(a\cos\theta,b\sin\theta)⋅(\cos\theta,\sin\theta)\}=a\cos^2\theta+b\sin^2\theta$.

Where is the error?

Answer 1

The main problem you have is giving the same name to different things: $\theta$ means two different things, and so does $a$.

The parametric equation $x=a\cos t$, $y=b\sin t$, leads to $$ h_E(\theta)=\sup_t (a\cos t,b\sin t)\cdot (\cos\theta, \sin\theta) $$ Observe that the dot product can just as well be written as $$ (\cos t,\sin t)\cdot (a\cos\theta, b\sin\theta) $$ which is simply the projection of $(a\cos\theta, b\sin\theta)$ onto the direction determined by $t$. The maximal possible value of scalar projection is the length of the vector, hence $$ h_E(\theta)=|(a\cos\theta, b\sin\theta)| = \sqrt{a^2\cos^2\theta+b^2\sin^2\theta} $$

Answer 2

A general hyper-ellipoid is the affine image of the unit-ball. This can be written as $\mathcal E := \{Ax + c \text{ s.t } x \in \mathbb B_n\}$, where $\mathbb B_n := \{x \in \mathbb R^n \text{ s.t } \|x\|_2 \le 1\}$ is the unit-ball in $\mathbb R^n$, $A : \mathbb R^n \rightarrow \mathbb R^n$ is a linear transformation and $c \in \mathbb R^n$, is the center of the ellipsoid. Now, one computes the support function $\sigma_{\mathcal E}$ of $\mathcal E$ as

\begin{equation} \begin{split} \sigma_{\mathcal E}(z) &= \sup\{\langle z, y\rangle \text{ s.t } y \in \mathcal E\} = \sup\{\langle z, Ax + c\rangle \text{ s.t } x \in \mathbb B_n\} \\ &= \langle z, c\rangle + \sup\{\langle A^Tz, x\rangle \text{ s.t } x \in \mathbb B_n\} = \langle z, c\rangle + \sigma_{\mathbb B_n}(A^Tz) = \langle z, c\rangle + \|A^Tz\|_{(2^*)}\\ &= \langle z, c\rangle + \|A^Tz\|_2. \end{split} \end{equation}

In your particular case $c = 0 \in \mathbb R^n$, and $A$ is an orthonormal matrix with entries $\pm \sin(\theta), \pm \cos(\theta)$. Figure out the the precise values for these entries, plug them into the formula i derived above, and you're done.