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Lydia Carey 2022-06-24 Answered
Suppose that p 1 , , p n are nonnegative real numbers such that p 1 + + p n = 1; denote the corresponding set of vectors by Δ n .

I am interested in the following function, f : Δ n R + , given by
f ( p ) = k = 1 n p k j = k n p j .
We always have f ( p ) n, using the lower bound j k p j p k . However I feel this must be a loose bound on the quantity
sup p Δ n f ( p ) ,
since it requires that j > k p j = 0 for all k to be met with equality. Hence, I am wondering what the largest f ( p ) can be when evaluated over the simplex?
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Answers (1)

kuncwadi17
Answered 2022-06-25 Author has 16 answers
Take 0 < r < 1 and put p k = ( r k 1 r k ) / ( 1 r n ) for 1 k n. Then
k = 1 n p k j = k n p j = j = 1 n r k 1 r k r k 1 r n = [ j = n k + 1 ] j = 1 n 1 r 1 r j .
This tends to n as r 0, thus sup p Δ n f ( p ) = n.
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