PreprintCC BY 4.0arXiv
Failure of Rockafellar's Sum Conjecture via an Explicit Non-(FPV) Operator
Abstract
Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on c0(N0×N). The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.
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BibTeX
@misc{bot2026failure,
title = {{Failure of Rockafellar's Sum Conjecture via an Explicit Non-(FPV) Operator}},
author = {Radu Ioan Bot},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13906},
eprint = {2609.13906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13906},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Failure of Rockafellar's Sum Conjecture via an Explicit Non-(FPV) Operator
AU - Radu Ioan Bot
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13906
UR - https://arxiv.org/abs/2609.13906
AB - Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on c_0(N_0×N). The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.
ER -