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Oct 1, 2020 · Your proof is correct, but a bit redundant. You showed that max(A) + max(B) ≥ x max (A) + max (B) ≥ x for all x ∈ A + B x ∈ A + B, and you explained how max(A) + max(B) ∈ A + B max (A) + max (B) ∈ A + B. This suffices to show what we need, as we've found an upper bound for A + B A + B that also belongs in A + B A + B, hence it's the ...
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$\begingroup$ I feel like allowing $\arg\max f(x)$ to be either $\in \mathbb{R}$ or $\in \mathcal{P}(\mathbb{R})$ is a very troublesome definition.
Aug 21, 2011 · $\max\{x_1,x_2\} = \cases{x_1, \text{if }x_1 > x_2\\x_2, \text{otherwise}}$ You can define like that the maximum of any finitely many elements. When the parameters are an infinite set of values, then it is implied that one of them is maximal (namely that there is a greatest one, unlike the set $\{-\frac{1}{n} | n\in\mathbb{N}\}$ where there is no greatest element)
$\begingroup$ I prefer $\max\{f(x_1,\ldots,f(x_n)\}$ with curly braces and no parentheses. In this instance, the parentheses don't actually help, and the curly braces remind you that the thing whose maximum is sought is a set rather than a tuple. $\endgroup$
Get the range of the required distribution, in this case, max(X, Y) Find the CDF of this distribution as a function of the known distributions Find the PDF of the distribution by differentiating the CDF
But let's take x = 2, then (1 - 2) ^ 2 will be (-1) ^2 which is nothing but 1 and according to op's max function, 1 should be returned. But since you gave the condition of x >= 1, we always return 0 even when x is something like 2. I think in comments what Andre Holzner said is correct.
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Sep 19, 2017 · One line proof: Since composition of convex functions is convex, we only need to show max (x, y) is convex. But max (x, y) = x + y 2 + | x − y 2 | and | ⋅ | is obviously convex. A function f: Rn → R is convex if and only if its epigraph epif = {(x, t) ∈ Rn × R ∣ f(x) ≤ t} is a convex set.
A general result called Von Neumann-Fan minimax theorem states the following: Theorem 2 (Von Neumann-Fan minimax theorem). Let X and Y be Banach spaces.Let C ⊂ X be nonempty and convex, and let D ⊂ Y be nonempty, weakly compact and convex. Let g: X × Y → R be convex with respect to x ∈ C and concave and upper-semicontinuous with ...
