Let $X$ be a real Banach space. Consider the direct sum $X\oplus X\oplus X$ with the norm $$\|(x,y,z)\|_1=\|x\|+\|y\|+\|z\|\text{ for all }x,y,z\in X.$$ I want to show that this space is not isometrically isomorphic to $X^*\oplus X^*\oplus X^*$ with the norm $$\|(x^*,y^*,z^*)\|_1=\|x^*\|+\|y^*\|+\|z^*\|\text{ for all }x^*,y^*,z^*\in X^*.$$
To answer this question, I want to know about the properties shared by isometrically isomorphic spaces. I know that the following properties are shared by two isometrically isomorphic spaces (there may be many more that I don't know):
- Extreme points
- Distance between subsets
together with the properties like completeness and other preserved by an isomorphism.
In this problem, I feel, it is possible to show that the closed unit ball of one of the spaces contains extreme points, but not the other. However I could not make any progress. Any hint is appreciated.