Abstract
The Demyanov–Ryabova conjecture is a geometric problem originating from duality relations between nonconvex objects. Given a finite collection of polytopes, one obtains its dual collection as convex hulls of the maximal facet of sets in the original collection, for each direction in the space (thus constructing upper convex representations of positively homogeneous functions from lower ones and, vice versa, via Minkowski duality). It is conjectured that an iterative application of this conversion procedure to finite families of polytopes results in a cycle of length at most two. We prove a special case of the conjecture assuming an affine independence condition on the vertices of polytopes in the collection. We also obtain a purely combinatorial reformulation of the conjecture.
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Acknowledgements
I would like to thank Vera Roshchina (RMIT University) for introducing this open problem to me. I also appreciate many helpful suggestions and support from David Yost (Federation University of Australia), Andrew Eberhard (RMIT University), and the anonymous referee, who reviewed my paper. This work was partially funded by the Australian government through the Australian Research Council, Project DE150100240.
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Communicated by Aris Daniilidis.
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Sang, T. On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters. J Optim Theory Appl 174, 712–727 (2017). https://doi.org/10.1007/s10957-017-1141-0
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DOI: https://doi.org/10.1007/s10957-017-1141-0