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If you have a bound on the uniform norm, say $\|f\|_{\infty, C}\le M$, the sequence has a uniform Lipschitz estimate $ 2M/r$ on the set $C_r \subset C$ of all points with distance at least $r$ …

Just out of curiosity, we may characterize the convex functions $\Phi:\mathbb{R}_+\to\mathbb{R}$ satisfying that condition as those convex functions that are affine on the interval $[1,\frac32]$ and …

You can have any pair $-\infty\le a\le b\le+\infty$ as limit inferior and limit superior, choosing a suitable concave function $g$ (and whatever is the nonnegative concave function $h(x)$ in place of …

Here is a quick application of the Ekeland's Variational Principle to Spectral Theory. Let $A$ be a bounded linear symmetric operator on a Hilbert space $H$, and let $\mathbb{S}$ be the unit sphere o …

An idea of the proof. For the convexity of $F$: a function is convex iff its epigraph is convex; the epigraph of $F$ is the projection of the epigraph of $f$; the projection of a convex set is convex. …

There could be an obstruction to continuity due to a lack of uniqueness of the representation of $o$ as convex combination. For example define $\Gamma_t$ to be, for $t\in\mathbb R$, the $\vee$-shape …

Denoting $f'$ either the right or the left derivative, for $0<x<y$ we have $\displaystyle f'(x)\le\frac{f(y)-f(x)}{y-x} \le f'(y)$, so $\displaystyle {f(y)-f(x)} \le y f'(y)-xf'(y)\le y f'(y)-xf'(x)$ …