PreprintCC BY 4.0arXiv
A Projection Identity for Simplices Sharp Inequalities, Converse Results, and Affine Projections
Abstract
We study a projection identity for a simplex in Euclidean space, written in terms of the frame operator of its unit edge directions. For a right simplex, the identity leads to a sharp family of distance inequalities and a complete description of equality. For a general simplex, the same formula is controlled by the spectrum of the Gram matrix through the Ky Fan principle. We prove converse results that characterise right simplices and determine the smallest number of projection subspaces needed to force orthogonality, together with an optimal quantitative estimate. We also treat affine projection subspaces and show how the original inequality for mutually perpendicular vectors fits into the same framework.
Related
- A Continuous Projection Converse and a Weighted Construction for Uniquely Decodable Code Pairs
- Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
- Stability of Hyperplane Projections of the Regular Simplex and Lp-Projection Bodies of balls in ℓqn(ℝ)
- Oblique Projection Networks (OPN) Reference guide - variants, code, tests, complexity, and the application scope of oblique projections
- Sobolev inequalities for sections and projections of log-concave functions
BibTeX
@misc{tran2026projection,
title = {{A Projection Identity for Simplices Sharp Inequalities, Converse Results, and Affine Projections}},
author = {Quang Hung Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01226},
eprint = {2609.01226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01226},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Projection Identity for Simplices Sharp Inequalities, Converse Results, and Affine Projections
AU - Quang Hung Tran
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01226
UR - https://arxiv.org/abs/2609.01226
AB - We study a projection identity for a simplex in Euclidean space, written in terms of the frame operator of its unit edge directions. For a right simplex, the identity leads to a sharp family of distance inequalities and a complete description of equality. For a general simplex, the same formula is controlled by the spectrum of the Gram matrix through the Ky Fan principle. We prove converse results that characterise right simplices and determine the smallest number of projection subspaces needed to force orthogonality, together with an optimal quantitative estimate. We also treat affine projection subspaces and show how the original inequality for mutually perpendicular vectors fits into the same framework.
ER -