If we follow the mean-risk ptf optimization strategy and use variance/VaR/ES as risk measure, the results in general are different. In other words the efficient frontier are not shared. However, as already said by Enrico Schumann, if the multivariate returns distribution of the assets is Elliptical (with finite variance) the frontier is shared. Note that the class of Elliptical distributions include the Normal one but is much more general, among others include t-student distribution.
The graph above can make sense only in the elliptical case, however It seems me strange yet.
In elliptical case entire frontier is shared and the global minimum risk point as well. In the graph are indicated three separated points/portfolios on the frontier; what minimum VaR/ES points mean? These two point not make so sense. These can be labeled as min mean variance too. Indeed in the horizontal line standard deviation is used. At the same points even min VaR and/or ES are achieved but the graph appear missleading to me.
Moreover in the source, before the graph, it is said that:
In this chapter we formulate and solve the mean-CVaR portfolio model,
where covariance risk is now replaced by the conditional Value at Risk
as the risk measure. In contrast to the mean-variance portfolio
optimization problem, we no longer assume the restriction consisting
in the set of assets to have a multivariate elliptically contoured
In this case is almost impossible that mean-VaR and/or mean-ES solutions are shared with mean-variance one; mean-VaR and/or mean-ES points must stay below the mean-variance frontier. Then I conclude that the graph is wrong.