"We often forget that growth in wealth approximately equals the arithmetic average return over an investment horizon less half the volatility of the return. This is a fundamental, indisputable mathematical principle"

Growth in its simplest meaning must be a ratio between ending and beginning wealth, this end/beginning ratio by definition do not care what happened in between.

The author may be referring to the arithmetic drag, which happen because something that goes up and down by the same ratio (e.g., +-20%) do not end up in its original place. The reason being each time a ratio is applied, the basis of the percentage calculation is changed. The arithmetic mean is a familiar tool but it is not a suitable tool to calculate return. If it must be used, then you will need to take account of the the above error, the volatility drag, which can be approximated by variance/2, such that, (arithmetic mean return) - (variance
of return)/2 ≈ geometric mean return, which is the above described ratio.

A higher variance however will not produce a lower geometric return. If we stick with the begin/end definition of return.

That being said, Shannon has demonstrated how volatility can be harvested systemically using what's later called CPPI in the 1960s, using a similar coin toss example as described above. It subsequently inspired much study in this area of portfolio selection. In my study of the above subject, I found this site. I hope this may help anyone who stumble on this in a search.