PreprintCC BY 4.0arXiv
Self-dual sets up to a translation: a negative answer to Milman's question
Abstract
Let K,T⊂ℝn be convex bodies with the origin in their interiors. If K=T∘, then the identity K+T=K∘+T∘ holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether K+T=K∘+T∘ forces K=T∘. We show that the answer is positive for polytopes but negative for general convex bodies.
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BibTeX
@misc{segal2026selfdual,
title = {{Self-dual sets up to a translation: a negative answer to Milman's question}},
author = {Alex Segal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12685},
eprint = {2609.12685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-dual sets up to a translation: a negative answer to Milman's question
AU - Alex Segal
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12685
UR - https://arxiv.org/abs/2609.12685
AB - Let K,T⊂ℝ^n be convex bodies with the origin in their interiors. If K=T^∘, then the identity K+T=K^∘+T^∘ holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether K+T=K^∘+T^∘ forces K=T^∘. We show that the answer is positive for polytopes but negative for general convex bodies.
ER -