# Questions tagged [convex-analysis]

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### Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?

I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13. In usul's question, the answer ...
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### subgradient in a predual under weak* continuity

Let $X$ be a Banach space. Suppose $f:X^*\to\mathbb R\cup\{\infty\}$ is convex, has closed and bounded (and so weak*-compact) effective domain, and is weak*-continuous on its effective domain. In ...
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### How are the $L^2$ and $\sup$ norms related on the space of strongly convex functions?

Given a convex compact set $X \subset \mathbb R^d$ with interior containing the orign let $V$ be the space of all smooth functions $f: X \to \mathbb R$ with the properties: The function is strongly ...
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### Characterization of global subdifferentiability

Let $X$ be a locally convex space, $D \subseteq X$ a nonempty compact convex set, and $f: D\to\mathbb R$ a continuous convex function. Question: Is there any known, interesting, alternative ...
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### Most general form of Jensen's inequality

What is the most general form of Jensen's inequality? Wikipedia gives for example this more general form, which holds in every topological vector space. Are there even more general forms, for ...
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### Extreme points of set of measures with given barycenter

Let $X$ be a convex compact metrizable subspace of a locally convex Hausdorff topological vector space, $x_0\in X$, and $P$ be the space of all Borel probability measures on $X$ with barycenter $x_0$. ...
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### Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
### Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article
The following is the multi-valued function I want to minimize: $f_1:\mathbb{R}^2\rightarrow \mathbb{R}$ defined by: $$f_1(x_1,x_2)=100x_1^2,\tag{1}$$ for all $x=(x_1,x_2)\in \mathbb{R}^2.$ The ...