I am very happy to announce that Jonas Wallin (Department of Statistics, Lund University), Barbara Wohlmtuh (Chair for Numerical Mathematics, TUM), and I put a new article on theory for active subspaces (ASM) on arXiv. ASM is already known to be applicable to settings involving probability distributions with compact support or of Gaussian-type. We investigate ASM in the context of distributions with exponential tails and were able to show that existing bounds are not valid anymore due to arbitrarily large Poincaré constants. Also, we propose a way for getting weaker, or generalized, bounds that, however, result in a lower order of the error bound. Indeed, we show how to balance the size the Poincaré constant and the order of the error. At the end, we suggest an open problem to the community which aims at generalizing the applicability of ASM to a larger class of distributions, i.e., multivariate generalized hyperbolics. I want to thank both of my co-authors for their feedback and assistance in developing the structure of this manuscript and for the valuable discussions we had.