Convex-Cyclicity and Convex Hulls of Orbits
DOI:
https://doi.org/10.53573/rhimrj.2026.v13n04.009Keywords:
Convex-cyclicity, subspace convex-cyclicity, orbitope, convex hull of an orbit, hypercyclicity, Banach space, cyclic operatorAbstract
Subspace convex-cyclicity generalizes the concept of convex-cyclicity, introduced by Rezaei in 2013, by requiring that the convex hull of an operator’s orbit be dense only in a specified closed subspace rather than the entire Banach space. This framework allows for a refined analysis of linear operators, particularly when global convex-cyclicity fails. Key theoretical properties include a Hahn-Banach characterization, relationships with subspace cyclicity, and the allowance of non-invariant subspaces. The convex hull of an orbit, known as an orbitope, forms highly symmetric polytopes or convex bodies under group actions, with fundamental applications in optimization and geometry. This work synthesizes operator-theoretic notions of convex-cyclicity with computational and geometric insights into convex hulls of orbits.
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