We show that ℓp norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the Lp norms are the unique rearrangement-invariant norms on a probability space such that ‖XY‖ = ‖X‖ ⋅ ‖Y‖ for every pair X, Y of independent random variables. Our proof combines the tensor power trick and Cramér's large deviation theorem.
Guillaume Aubrun 1 ; Ion Nechita 1
@article{CML_2011__3_4_637_0,
author = {Guillaume Aubrun and Ion Nechita},
title = {The multiplicative property characterizes $\ell _p$ and $L_p$ norms},
journal = {Confluentes Mathematici},
pages = {637--647},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {4},
year = {2011},
doi = {10.1142/S1793744211000485},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/}
}
TY - JOUR AU - Guillaume Aubrun AU - Ion Nechita TI - The multiplicative property characterizes $\ell _p$ and $L_p$ norms JO - Confluentes Mathematici PY - 2011 SP - 637 EP - 647 VL - 3 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/ DO - 10.1142/S1793744211000485 LA - en ID - CML_2011__3_4_637_0 ER -
%0 Journal Article %A Guillaume Aubrun %A Ion Nechita %T The multiplicative property characterizes $\ell _p$ and $L_p$ norms %J Confluentes Mathematici %D 2011 %P 637-647 %V 3 %N 4 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/ %R 10.1142/S1793744211000485 %G en %F CML_2011__3_4_637_0
Guillaume Aubrun; Ion Nechita. The multiplicative property characterizes $\ell _p$ and $L_p$ norms. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 637-647. doi : 10.1142/S1793744211000485. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/
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