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I need to know which norms can be consistently decomposed in the sense that if I take a vector $v = (a,b,c)$ and build a vector $v' = (\|(a,b)\|,c)$ we have $\|v\| = \|v'\|$.

More precisely, let $v = \sum_{i=1}^n v_ie_i$ be a finite-dimensional vector, and $\{P_j\}_{j=1}^k$ a partition of the index set $\{i\}_{i=1}^n$ into $k$ subsets, so that $v = \sum_{j=1}^k \sum_{i \in P_j} v_ie_i$. The question is then for which norms is it true that for all vectors and partitions $$ \|v\| = \left\|\sum_{j=1}^k \Bigg\|\sum_{i \in P_j} v_ie_i\Bigg\|e_j\right\|. $$

It is easy to see that this is true for every $p$-norm, and every other norm that I've tried failed to have this property, so it would be natural to conjecture that $p$-norms are the only consistently decomposable ones. Just finding a counterexample to this conjecture would be very useful.

I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that $$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$

More precisely, let $v = \sum_{i=1}^n v_ie_i$ be a finite-dimensional vector, and $\{P_j\}_{j=1}^k$ a partition of the index set $\{i\}_{i=1}^n$ into $k$ subsets, so that $v = \sum_{j=1}^k \sum_{i \in P_j} v_ie_i$. The question is then for which norms is it true that for all vectors and partitions $$ \|v\| = \left\|\sum_{j=1}^k \Bigg\|\sum_{i \in P_j} v_ie_i\Bigg\|e_j\right\|. $$

It is easy to see that this is true for every $p$-norm, and every other norm that I've tried failed to have this property, so it would be natural to conjecture that $p$-norms are the only consistently decomposable ones. Just finding a counterexample to this conjecture would be very useful.