scholarly journals Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points

2002 ◽  
Vol 115 (1) ◽  
pp. 44-55 ◽  
Author(s):  
Chong Li ◽  
Renxing Ni
Keyword(s):  
Distance Functions ◽  
Derivatives Of ◽  
Nearest Points ◽  
Generalized Distance

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

Quality, Valuation, and Welfare

2020 ◽  
pp. 303-354
Author(s):  
Robert G. Chambers
Keyword(s):  
Minimum Principle ◽  
Consumer Welfare ◽  
Production Quality ◽  
Distance Functions ◽  
Production Model ◽  
Directional Derivatives ◽  
Analytic Structure ◽  
Compensating Variation ◽  
Nontraded Goods ◽  
Derivatives Of

The analytic structure developed in the first six chapters is applied to quality-differentiated production, quality-differentiated pricing, and consumer welfare analysis. The quality-differentiated production problem is developed as a special case of the multiple-output problem for both nonstochastic and stochastic pricing regimes. The "household production" model of Gorman (1956) and Lancaster (1966) is developed in a conjugate dual framework whose solution for rational individuals obeys the zero-maximum (zero minimum) principle. The nominal concepts of compensating variation and equivalent variation are shown to have real-valued (dual) parallels in the compensating benefit and the equivalent benefit. Real, as opposed to nominal, valuation for traded and nontraded goods is treated in the benefit framework. Directional derivatives of distance functions are used to rationalize the frequently observed empirical discrepancy between willingness-to-pay and willingness-to-accept.

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