# On convex functions which are non constant on every segment

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.

• To the down-voter: could you please leave a constructive feedback so that I can modify my question accordingly? Thanks.
– Y.B.
Commented Mar 11, 2019 at 19:35
• 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. Commented Mar 16, 2019 at 21:35

strictly convex means: convex and not affine on any non-trivial line segment.

So B is something between convex and strictly convex.

• 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.
– Y.B.
Commented Mar 14, 2019 at 15:11