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I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the variational properties of a class of funtionals - that they call demi-coercive. The definition is the following:

Definition. A function $f\colon \mathbb R^N \to \mathbb R$ is demi-coercive (DC) if there are $a>0, b \ge 0$ and $\gamma \in \mathbb R^N$ such that $$ a|x| \le f(x) + \langle \gamma, x \rangle + b $$ for any $x \in \mathbb R^N$.

Among other interesting characterizations (see Thm. 2.4 of the above paper) one is of particular interest to me as of today. Namely:

Prop. (A) A convex, lower semicontinuous function $f\colon \mathbb R^N \to [0,+\infty)$ is DC iff there are no straight lines along which $f$ is constant.

I am interested into this proposition for the following reason: I need to modify the above property by considering the more restrictive variant:

Prop/Def. (B) A convex, lower semicontinuous function $f\colon \mathbb R^N \to [0,+\infty)$ is ? iff there are no non-trivial straight segments along which $f$ is finite and constant.

Indeed, in Prop. (A) the equivalence relies really on the fact that there are no lines on which $f$ is constant (so it can be constant on a bounded segment, even on half lines).

On the contrary, in Prop. (B) I want to rule out the possibility that $f$ is constant even on a segment.

Question(s).

  1. What should I put in place of (?) ? I would like to know if Prop/Def (B) has ever been considered in literature (do these functions have a name?).

  2. I would also like to see if there are equivalent characterizations: in particular, is it possible to say something relating to (B) in terms of the level sets of $f$ and their extreme points?

Notice that easy examples of (1-homogeneous) functions satisfying A but not B are the $\ell^1$ or $\ell^\infty$ norm on $\mathbb R^N$, as one can directly check. Notice that for all others $p \in (1,+\infty)$ the $p$-norm does satisfy both A and B.

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    $\begingroup$ To the down-voter: could you please leave a constructive feedback so that I can modify my question accordingly? Thanks. $\endgroup$ – Y.B. Mar 11 at 19:35
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    $\begingroup$ I do not know the answer, but if there is no sexy name for this class, it could be called in a descriptive manner "a class of convex functions with strictly convex level sets". This is no shorter than "convex and not constant on any segment", though. $\endgroup$ – Mateusz Kwaśnicki Mar 16 at 21:35
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strictly convex means: convex and not affine on any non-trivial line segment.

LINK

So B is something between convex and strictly convex.

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    $\begingroup$ Yes exactly, the point is indeed to put precisely how B is "between" strictly convex and convex. As for A, if I am not wrong, it should hold something like this: A is the "closure" (in a suitable sense) of strictly convex functions (see e.g. this thesis, Thm. 3.5). I would like to get something like this for B... Thanks for your answer. $\endgroup$ – Y.B. Mar 14 at 15:11

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