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Support

Author

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  • Bruce C. Dieffenbach

    (Independent author)

Abstract

A proper, closed, convex function is positively homogeneous if and only if it is the support to a non-empty, closed, convex set. We interpret the Fenchel equality for the indicator and the support as a supporting hyperplane. The Cauchy-Schwarz inequality shows that the absolute value of the cosine of the angle between two vectors is less than or equal to one. In a statistical application in which the inner product is the covariance, the correlation between two random variables is less than or equal to one in absolute value. We study efficiency or weak efficiency by maximizing the efficiency function or the weak efficiency function. These two functions enable us to distinguish sharply and effectively between the two concepts. The maximization is especially fruitful in primal/dual convex optimization. The primal analyzes whether a particular vector of goals is efficient or weakly efficient, and this vector is the perturbation variable. One interprets the dual choice variable as a shadow price for the goals.

Suggested Citation

  • Bruce C. Dieffenbach, 2026. "Support," Contributions to Economics,, Springer.
    • Handle: RePEc:spr:conchp:978-3-032-21396-9_5
    DOI: 10.1007/978-3-032-21396-9_5
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