I have been studying for the last few weeks the paper [Dirichlet problem for demi-coercive functionals][1] 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.


  [1]: https://www.sciencedirect.com/science/article/pii/0362546X86901458