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Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains

  • Gabriel Raposo
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Abstract

We prove a generalized circular rearrangement inequality: among all circular arrangements of a finite collection of positive numbers, the greedy arrangement simultaneously maximizes the sums of products of k consecutive entries for every k. This resolves a 2023 conjecture of Holmes-Holroyd-Ramírez and extends the classical circular rearrangement inequality for products of adjacent entries. As an application, we show that the greedy ordering minimizes the speed of a cyclic birth-and-death chain, resolving another conjecture of the same authors.

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Cite

BibTeXRIS
BibTeX
@misc{raposo2026circular,
  title = {{Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains}},
  author = {Gabriel Raposo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11774},
  eprint = {2609.11774},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11774},
}
RIS
TY  - UNPB
TI  - Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains
AU  - Gabriel Raposo
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11774
UR  - https://arxiv.org/abs/2609.11774
AB  - We prove a generalized circular rearrangement inequality: among all circular arrangements of a finite collection of positive numbers, the greedy arrangement simultaneously maximizes the sums of products of k consecutive entries for every k. This resolves a 2023 conjecture of Holmes-Holroyd-Ramírez and extends the classical circular rearrangement inequality for products of adjacent entries. As an application, we show that the greedy ordering minimizes the speed of a cyclic birth-and-death chain, resolving another conjecture of the same authors.
ER  -