Let $\gamma$ be closed, smooth, strictly convex curve. It is known that we can find a special linear transformation $A\in SL(2)$ so that the length of $A\gamma$ is minimized.
I am interested in knowing of a test/ criteria $(\ast)$ that if a given closed, smooth, strictly convex curve $\gamma$ satisfies $(\ast)$, then its length must be already minimized; that is, we do not need to apply any special linear transformation to $\gamma$ to reach the minimum length.
In terms of the support function, I have the following necessary conditions that $\gamma$ must satisfy. Before stating the conditions, let me first define the support function of $\gamma.$ The support function of a smooth, strictly convex, closed curve $\gamma$ is defined by $h(\theta)=\langle x, (\cos \theta, \sin \theta) \rangle,$ where $x$ is the point on $\gamma$ with the outer unit normal $(\cos \theta, \sin \theta)$ and $\theta\in[0,2\pi]$ (It then follows that $x=(h\cos\theta -h_{\theta}\sin\theta,h\sin\theta+h_{\theta}\cos\theta)$ and thus we have a parametrization of $\gamma$ over the unit sphere $\mathbb{S}^1$ identified with $[0,2\pi]$.). It is also known that the length of $\gamma$ can be expressed as follows $$L(\gamma)=\int_{0}^{2\pi}h~d\theta.$$
Now the conditions: if $L(A\gamma),~A\in SL(2),$ is minimized for $A=I$, then
$$0=\int_{0}^{2\pi}D_{L_i}hd\theta,i=1,2~~~(\ast)$$ where $L_1=\left( \begin{array}{cc} 1 & 0\\ 0 & -1\\ \end{array} \right) $ and $L_2=\left( \begin{array}{cc} 0 & 1\\ 1 & 0\\ \end{array} \right) $. Note $L_1,L_2$ are the generatores of the Lie group of $SL(2).$ Here, I should say $$D_{L_i}h=\langle L_ix^T, (\cos \theta, \sin \theta)\rangle.$$
Doing some calculations $(\ast)$ implies that $\int_0^{2\pi}h\cos2\theta d\theta=\int_0^{2\pi}h\sin2\theta d\theta=0.$
Now my questions are:
1- given a curve $\gamma$ satisfying $(\ast)$, is it true that the length of $\gamma$ is already minimized?
2- generally, given $\gamma,$ can I explicitly find an $A\in SL(2)$ such that the length of $A\gamma$ is minimum?
3- what is the direct generalization to higher dimensions for hypersurfaces? (I know this last question is very vague)