# 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. Mar 11 '19 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. – Mateusz Kwaśnicki Mar 16 '19 at 21:35