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Register a new userA counterexample to De Pierros conjecture on the convergence of under-relaxed cyclic projections
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The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the $varepsilon$-under-relaxed cyclic projection method converge when $varepsilondownarrow 0$ towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in $mathbb{R}^3$ for which the $varepsilon$-under-relaxed cycles do not converge.
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We prove that the Laptev--Safronov conjecture (Comm. Math. Phys., 2009) is false in the range that is not covered by Franks positive result (Bull. Lond. Math. Soc., 2011). The simple counterexample is adaptable to a large class of Schrodinger type operators, for which we also prove new sharp upper bounds.
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