Suppose that we have a portfolio of $n$ assets.
A perfectly diversified portfolio is one in which each asset has equal weights, i.e. each asset has weight $\frac{1}{n}$. Of course this is usually not the case.
What are some of the ways we can measure how well diversified our portfolio is?
We could measure how far our portfolio is from the equally-weighted portfolio.
This of course will depend on the geometry of the space which is not euclidean since the sum of the weights must be one.
Answer
If you measure risk by the standard deviation of the portfolio return $$ \sigma = \sqrt{w^T \Sigma w}, $$ then it is usual to define risk contributions for each asset by $$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$ then diversified could mean that these $\sigma_i$ are evenly spread over the assets in the portfolio.
You find this approach and more in this paper by Meucci
There you also find the variance concentration curve that uses principle components (PCs) of the asset universe and the weighting of the assets to analyze how much the PCs contribute.
Ad good place to read about the application of PCA to portfolio analysis is Regularization of Portfolio Allocation by B. Bruder, N. Gaussel, J-C. Richard and T. Roncalli.
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