Stochastic forms of functional isoperimetric inequalities

The Brunn-Minkowski and Prekopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions, so it seems that convexity has no special significance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. In addition, a number of other geometric inequalities for convex sets arising from Brunn's concavity principle have recently been shown to yield local stochastic formulations, e.g., the Blaschke-Santalo inequality. In the first part of this dissertation, we study reverse inequalities for functionals of polar convex bodies invariant under the general linear group. We strengthen planar isoperimetric inequalities; we do that by attaching a stochastic model to some classical ones, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, for the polar of L[subscript p] centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Boroczky. For the rest of the dissertation, we initiate a systematic study of stochastic isoperimetric inequalities for random functions. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prekopa-Leindler inequality. Ultimately, we extend the latter result by developing stochastic geometry of s-concave functions. In this way, we establish local versions of dimensional forms of Brunn's principle a la Borell, Brascamp-Lieb, and Rinott. To do so, we define shadow systems of sconcave functions and revisit Rinott's approach in the context of multiple integral rearrangement inequalities.

The Impact of Shadow IT Systems on Perceived Information Credibility and Managerial Decision Making

SYNOPSIS Business trends show that more and more employees are creating shadow IT systems—IT systems that are not sanctioned or monitored by the IT department. This paper examines how the use of shadow IT in product costing impacts managers' perceptions of information credibility and managerial decision making. Using two experiments, we find that participants view information from shadow IT systems as less credible and they are less impacted by and less willing to rely on costing reports produced from shadow IT systems versus non-shadow IT systems. We also find that although participants are concerned about the credibility of shadow IT systems, they are not more likely to find simple mathematical errors embedded in shadow IT costing reports relative to non-shadow IT reports. This suggests that although concerned about shadow IT systems, managers still do not exercise sufficient care in evaluating reports created using these systems. The results of our study should prove informative as shadow systems become more prevalent in organizations. Data Availability: Contact the authors.