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Mock 2018 QCM Regularization term

Can you give me an example of an objective where we had a regularization term in order to render the minimization problem strictly convex/concave? :/ I am not convinced.
Thank you for your help!
Screenshot 2022-01-14 at 09.06.06.jpg

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Take linear regression for example, the hessian is given by \(XX^\top /N\) for \(X\in \mathcal{R}^{d\times N}\).
If this matrix is not full rank then the problem is not strictly convex and in particular we can't invert the matrix \(XX^\top\), but with ridge regression we add a \(\lambda\) to all directions and thus ensure that the smallest eigenvalue of the hessian is at least this \(\lambda\), thus we have made the problem strictly convex.

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