This problem comes from the response of the author of papers.

Consider two convex bodies $A$ and $B$:

$$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$

$$B = \operatorname{conv} SO(3)$$

- $\mathcal{S}^4$ is the set of symmetric $4\times 4$ matrices.
- $A$ is a $9$ dimensional convex body. ($A$ is symmetric, so $10$ dimensional body, and $\operatorname{tr}(X) = 1$ will decrease one degree of freedom in the diagonal. Just imagine the vectorization of a matrix.) Of course, the extreme points of $A$ are those matrices in $A$ with rank one.
- $B$ is also a $9$ dimensional convex body (convex hull of rotation matrices).

I have two questions:

- What is the difference between "affinely isomorphic" and "isomorphic"? I try to search some lectures; however, I cannot fully understand it still. Hope for a plain explanation.
- How to prove that both convex bodies are affinely isomorphic?