What does "a convex-$\bigcap$ continuation" mean?

Question

I'm reading a paper named The Context-Tree Weighting Method Basic Properties, 1995 and everyone can access to http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.14.352&rep=rep1&type=pdf for free.

I have a question about a math symbol in this paper. It is at the beginning of Part B of Chapter 6, page 5.

Definition 8: Let $$\begin{eqnarray}\gamma(z)= \begin{cases} z, &0 \le z < 1 \cr \frac{1}{2}\log z + 1, & z \ge 1 \cr \end{cases} \end{eqnarray}$$ hence $\gamma(.)$ is a convex-$\bigcap$ continuation of $\frac{1}{2}\log z + 1$ for $0 \le z < 1$ satisfying $\gamma(0) = 0$.

Does anyone know what "a convex-$\bigcap$ continuation of" means? I never seen this in mathematics before.

Update:

Initially, I guess $\bigcap$ means the shape of $\gamma()$ coz if I draw it out in 2-D plane(X-Y), the shape of $\gamma$ will look like an upside-down bowl. But it is not a convex function. Besides, it is a little bit funny that a celebrated paper did use such an unprofessional symbol.

Answer 1

The standard terminology is "convex vs concave", but there are variations (indeed the names are confusing). For example:

      convex        |     concave 
  convex downwards  |  convex upwards
    concave up      |    convex cap

The "cap" sign, in convex-$\bigcap$, points surely to the latest - and that terminology is not too rare - and, granted, is the less ambiguous. It's what normally we call a concave function.

In your example, it's not strictly concave.