1,259 research outputs found
Polynomial Norms
In this paper, we study polynomial norms, i.e. norms that are the
root of a degree- homogeneous polynomial . We first show
that a necessary and sufficient condition for to be a norm is for
to be strictly convex, or equivalently, convex and positive definite. Though
not all norms come from roots of polynomials, we prove that any
norm can be approximated arbitrarily well by a polynomial norm. We then
investigate the computational problem of testing whether a form gives a
polynomial norm. We show that this problem is strongly NP-hard already when the
degree of the form is 4, but can always be answered by testing feasibility of a
semidefinite program (of possibly large size). We further study the problem of
optimizing over the set of polynomial norms using semidefinite programming. To
do this, we introduce the notion of r-sos-convexity and extend a result of
Reznick on sum of squares representation of positive definite forms to positive
definite biforms. We conclude with some applications of polynomial norms to
statistics and dynamical systems
Pareto efficiency for the concave order and multivariate comonotonicity
In this paper, we focus on efficient risk-sharing rules for the concave
dominance order. For a univariate risk, it follows from a comonotone dominance
principle, due to Landsberger and Meilijson [25], that efficiency is
characterized by a comonotonicity condition. The goal of this paper is to
generalize the comonotone dominance principle as well as the equivalence
between efficiency and comonotonicity to the multi-dimensional case. The
multivariate setting is more involved (in particular because there is no
immediate extension of the notion of comonotonicity) and we address it using
techniques from convex duality and optimal transportation
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