# Tagged Questions

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### Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
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### Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
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### Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
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### Show $f(a)=\operatorname{E} \min(aX,Y)$ is a quasilinear function of $a$
Let $X$ and $Y$ be bounded real-valued random variables. Define $$f(a)=\operatorname{E} \min(aX,Y)$$ Is $f$ a quasilinear function of $a$? That is $f$ is both quasiconvex and quasiconcave. To ...
Let $f(x)$ be a smooth function from $\mathbb{R}^n_{+}$ to $\mathbb{R}_{+}$ (it may be analytic). Are there some sufficient conditions on such function $f$ such that  f(x_1 y_1, \ldots, x_n y_n) ...