# Is the exponent in the rectangular matrix multiplication convex?

My question is regarding the paper "Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor". In the paper, the authors show an algorithm for multiplying a $$n \times n^c$$ and a $$n^c \times n$$ matrix. The authors show a graph of how the exponent in the time complexity depends on $$c$$. This graph is clearly convex. This would mean that one can upper bound the time complexity by linearly interpolating the exponent between that case of $$c = 0.31389$$ (where the exponent is 2) and $$c=1$$ (where the exponent is $$\omega$$). This would be useful for stating the time complexity of an algorithm I came up with.

However, they do not prove that the graph is convex. Is it convex? If so, any ideas as to how to cite this fact?

https://arxiv.org/abs/1708.05622