Consider a setting with n sellers having i.i.d. costs with log-concave density f from cumulative F , and a buyer who puts a premium i on procuring from seller i. We show how for any given 1; ; n; a simple second price bonus auction can be chosen which comes surprisingly close to giving the auctioneer the same surplus as an optimal mechanism. The bonuses depend only on the magnitude and monotoni...

In this paper, we prove that the extended values E(r, s;x, y) are Schur harmonic convex (or concave, respectively) with respect to (x, y) ∈ (0,∞) × (0,∞) if and only if (r, s) ∈ {(r, s) : s ≥ −1, s ≥ r, s+ r + 3 ≥ 0} ∪ {(r, s) : r ≥ −1, r ≥ s, s+r+3 ≥ 0} (or {(r, s) : s ≤ −1, r ≤ −1, s+r+3 ≤ 0}, respectively).

We present an extension of Fenchel’s duality theorem to nearly convexity, giving weaker conditions under which it takes place. Instead of minimizing the difference between a convex and a concave function, we minimize the subtraction of a nearly concave function from a nearly convex one. The assertion in the special case of Fenchel’s duality theorem that consists in minimizing the difference bet...

We discuss the relationship between matroid rank functions and a concept of discrete concavity called M-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are M-concave functions, while the (weighted) sum of matroid rank functions is not M-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions ...

We establish a combinatorial connection between the sequence (in,k) counting the involutions on n letters with k descents and the sequence (an,k) enumerating the semistandard Young tableaux on n cells with k symbols. This allows us to show that the sequences (in,k) are not log-concave for some values of n, hence answering a conjecture due to F. Brenti.

We present different notions of convexity and concavity for copulas and we study the relationships among them.

Let Vn(q) be the n-dimensional vector space over the finite field with q elements, and let T1, T2, . . . , Tr+1 be r+1 subspaces of Vn(q) such that Vn(q) = ⊕r+1 i=1 Ti. Assuming that dim(Ti) = 2 for 1 ≤ i ≤ r, let F be the collection of all subspaces S’s of Vn(q) such that S = ⊕r+1 i=1 (S ∩ Ti) and S contains at least one Ti for some 1 ≤ i ≤ r. In this paper we will prove that F is log concave ...

If & ^ 1 for each v, then by reasoning analogous to that of the preceding example, it may be shown, for any set (a), that there is no point p such that tp implies that log St(a, £) is concave. Hence Theorem 4 applies to all such functions log St(a, £). However, for this case the conclusion of the general theorem is weaker than the...

Let P denote the collection of positive sequences defined on N. Fix w ∈ P. Let s, t, respectively, be the sequences of partial sums of the infinite series ∑ wk and ∑ sk, respectively. Given x ∈ P, define the sequences A(x) and G(x) of weighted arithmetic and geometric means of x by An(x) = n ∑ k=1 wk sn xk, Gn(x) = n ∏ k=1 x wk/sn k , n = 1, 2, . . . Under the assumption that log t is concave, ...