Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known (see for example here: Embedding of Lp spaces), that for $ 1\leq p < q \leq +\infty$ we have an inclusion (embedding) $$ T^q_p \ : \ L^q([0,1],\lambda) \rightarrow L^p([0,1], \lambda),$$ which is linear (thus also affine).
One thing I know about affine transformations of real $\mathbb{R}^{n}$ spaces is, that it can transform an ellipsoid into a sphere (and vice versa). This is why I wonder what can we say about this fact in the more abstract setting of $L^p$ spaces. Let as take $p=2$ and any $q>2$. Define $$S^q_2=\{f\in L^q:||f||_2=1\}.$$ Then $S_2^q$ considered in the $L^2$ space is a subset of the unit sphere. The question is
As $T_2^q(S_2^q)=S_2^q$, and $T_2^q$ is affine, does it hold true, that $S_{2}^q$ is "some kind" of an ellipsoid in the $L^q$ space?
If so, how is an appropriate notion of such ellipsoid defined? The only thing that comes to my mind would possibly be as a set of points such that a sum of their distances from some given $f$ and $g$ functions is fixed. But this is only a childish guess... I would be glad for any insight.