What are the similarities and differences among global minimum variance (GMV), risk parity (RP), and low beta portfolios? Why might these portfolios outperform (as measured by Sharpe ratio) when they clearly do not have the maximal Sharpe Ratio in theory? Why might be some of the reasons for an investment consultant to not recommend these portfolios constructed from US equities given their seemingly superb performance characteristics when compared to the S&P 500?

Don't answer the question above it is needed to answer the question below.

Following up Question 3 above, under what condition will GMV and risk parity be tangency portfolio? You must show mathematical proof