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.
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?).
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.