PreprintCC BY 4.0arXiv
Finite-dimensional normed spaces without small-norm hyperplane projections
Abstract
We prove that there exists an absolute constant c>1 such that for every n ≥ 3 there exists an n-dimensional real normed space X for which every linear projection P onto a hyperplane satisfies |P| ≥ c. More generally, for every integer k ≥ 1 there exists a constant ck>1 such that for every n ≥ k+2 there exists an n-dimensional real normed space X such that every projection P onto a subspace of codimension at most k satisfies |P| ≥ ck. The proof is based on an idea of Gleit and Lazar, who established an analogous result for the space of real n × n matrices equipped with either the trace-class norm or the operator norm
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BibTeX
@misc{kobos2026finitedimensional,
title = {{Finite-dimensional normed spaces without small-norm hyperplane projections}},
author = {Tomasz Kobos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09113},
eprint = {2610.09113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09113},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-dimensional normed spaces without small-norm hyperplane projections
AU - Tomasz Kobos
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09113
UR - https://arxiv.org/abs/2610.09113
AB - We prove that there exists an absolute constant c>1 such that for every n ≥ 3 there exists an n-dimensional real normed space X for which every linear projection P onto a hyperplane satisfies |P| ≥ c. More generally, for every integer k ≥ 1 there exists a constant c_k>1 such that for every n ≥ k+2 there exists an n-dimensional real normed space X such that every projection P onto a subspace of codimension at most k satisfies |P| ≥ c_k. The proof is based on an idea of Gleit and Lazar, who established an analogous result for the space of real n × n matrices equipped with either the trace-class norm or the operator norm
ER -