A Property of Convex Sets
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From: Maury Barbato on 4 Aug 2010 10:23 Hello, let C be a compact convex subset of the usual euclidean space R^n, and suppose that C has at least two points. Does there exist two distinct points x, y in C such that the linear function f(z) = z*(x-y) (where * denotes the dot product in R^n) attains its maximum on C in x and its minimum on C in y? Thank you very much for your attention. My Best Regards, Maury Barbato PS Note that my question has a precise geometric meaning, and my geometric intuition suggests me it has a positive answer.
From: Robert Israel on 4 Aug 2010 15:24 Maury Barbato <mauriziobarbato(a)aruba.it> writes: > Hello, > let C be a compact convex subset of the usual euclidean > space R^n, and suppose that C has at least two points. Does there exist > two distinct points x, y in C such that > the linear function f(z) = z*(x-y) (where * denotes the > dot product in R^n) attains its maximum on C in x and > its minimum on C in y? Yes. Consider the continuous function g defined on C x C by g(s,t) = |s-t| (where |.| is the Euclidean norm). By compactness, g attains its supremum, i.e. there are x,y in C such that |x-y| >= |s-t| for all s,t in C. Then for any s, t in C f(s) - f(t) = (s - t)*(x - y) <= |s - t| |x - y| <= |x - y|^2 = f(x) - f(y). In particular (taking t = y) f(s) <= f(x) and (taking s = x) f(t) >= f(y), so f attains its maximum at x and its minimum at y. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Maury Barbato on 4 Aug 2010 12:59
Robert Israel wrote: > Maury Barbato <mauriziobarbato(a)aruba.it> writes: > > > Hello, > > let C be a compact convex subset of the usual > euclidean > > space R^n, and suppose that C has at least two > points. Does there exist > > two distinct points x, y in C such that > > the linear function f(z) = z*(x-y) (where * denotes > the > > dot product in R^n) attains its maximum on C in x > and > > its minimum on C in y? > > Yes. Consider the continuous function g defined on C > x C by g(s,t) = |s-t| > (where |.| is the Euclidean norm). > By compactness, g attains its supremum, i.e. there > are x,y in C such that > |x-y| >= |s-t| for all s,t in C. Then for any s, t > in C > f(s) - f(t) = (s - t)*(x - y) <= |s - t| |x - y| <= > |x - y|^2 = f(x) - f(y). > In particular (taking t = y) f(s) <= f(x) and (taking > s = x) f(t) >= f(y), > so f attains its maximum at x and its minimum at y. > -- > Robert Israel > israel(a)math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, > BC, Canada Chapeau, monsieur le professeur! Your solutions are always clever and clear! Thank you very very ... much, prof. Israel! Friendly Regards, Maury Barbato |