PreprintCC BY 4.0arXiv
Monotone invariant valuations on convex cones
Abstract
We classify the monotone SOn d-invariant valuations, d≥2, on the space of all closed convex cones. They are precisely the linear combinations of the conic intrinsic volumes with nondecreasing coefficients, or equivalently, the nonnegative linear combinations of intrinsic volume tails, up to an additive constant. On nonzero pointed cones we obtain the corresponding characterisation by Grassmann angles, settling a conjecture of McMullen. Every such valuation is automatically continuous in the spherical Hausdorff topology and is On d-invariant. Our proof builds on the signed simplex and averaging arguments of Wang and Wu, but does not rely on the continuous classification theorem.
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BibTeX
@misc{lotz2026monotone,
title = {{Monotone invariant valuations on convex cones}},
author = {Martin Lotz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09335},
eprint = {2609.09335},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09335},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monotone invariant valuations on convex cones
AU - Martin Lotz
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.09335
UR - https://arxiv.org/abs/2609.09335
AB - We classify the monotone SOn d-invariant valuations, d≥2, on the space of all closed convex cones. They are precisely the linear combinations of the conic intrinsic volumes with nondecreasing coefficients, or equivalently, the nonnegative linear combinations of intrinsic volume tails, up to an additive constant. On nonzero pointed cones we obtain the corresponding characterisation by Grassmann angles, settling a conjecture of McMullen. Every such valuation is automatically continuous in the spherical Hausdorff topology and is On d-invariant. Our proof builds on the signed simplex and averaging arguments of Wang and Wu, but does not rely on the continuous classification theorem.
ER -