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In this paper the topological property of arcwise connectedness is generalized both for sets and functions: a family of “modified secants” is defined, not referring to particular kinds of functions, quadratic, logarithmic, exponential, etc., but fixing the extreme configurations of the generalized means. Such a theory is developed by proving properties which generalize convexity and quasi-convexity in a topological rather than a purely algebraic framework. A new separation theorem is given and a theory of the alternative is stated for the new functions introduced in order to unify the convex and quasi-convex results. © 1987 Taylor & Francis Group, LLC. All rights reserved.
Generalized connectedness for families of arcs
CASTAGNOLI, ERIO;
1987
Abstract
In this paper the topological property of arcwise connectedness is generalized both for sets and functions: a family of “modified secants” is defined, not referring to particular kinds of functions, quadratic, logarithmic, exponential, etc., but fixing the extreme configurations of the generalized means. Such a theory is developed by proving properties which generalize convexity and quasi-convexity in a topological rather than a purely algebraic framework. A new separation theorem is given and a theory of the alternative is stated for the new functions introduced in order to unify the convex and quasi-convex results. © 1987 Taylor & Francis Group, LLC. All rights reserved.
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