Isometric embedding of $\mathbb{E}^3$ into Finsler space

Question

Consider the following paper : Euclidean rank for Finsler Space

If $(\mathbb{R}^3,\|\ \|_i)$ are normed space, then there is an isometric embedding of $\mathbb{E}^3$ into $(\mathbb{R}^3,\|\ \|_1)\times (\mathbb{R}^3,\|\ \|_2)$ where $(\mathbb{R}^3,\|\ \|_i)$ has no isometric embedding of $\mathbb{E}^2$.

For the proof, they introduce the following exercise : When $S_0$ is 2-dimensional Euclidean sphere, define

$$ k(t):=\frac{1}{N}\ \sin\ (t+\pi/4)\sin\ (t-\pi/4)\sin\ (t+3\pi/8)\sin\ (t-3\pi/8)$$

$$ t=|x-u|,\ s=|x-v| $$ where $|u-v|=\frac{\pi}{2}$. Clearly, $$ S=\bigg\{ x(1+k(t)k(s) )\bigg| x\in S_0\bigg\} $$ is homeomorphic to a sphere when $N$ is large. (In further, any great circle in $S_0$ is not in $S$.)

Prove that $S$ can be a convex surface for some $N$.

Answer 1

I believe that the following exercise solve the question : Prove that $c(t)=(\cos\ t,\sin\ t)f(t),\ f(t)=(1+ \frac{\cos\ 2n t}{N})$ is convex for a large $N$.

Proof : When $c\circ t (u)$ is unit speed curve, then let $F(u) =\frac{f^2}{2}$ so that $F''(u)\leq 1$ for a large $N$. Hence $c$ is a convex curve.