Gee, it's fun to read about Arrow's work! From the OP, I see that I've been closer to Arrow's work than I knew! In grad school, I had a relatively severe intr

Gee, it's fun to read about Arrow's work! From the OP, I see that I've been closer to Arrow's work than I knew! In grad school, I had a relatively severe introduction to optimization including both linear and non-linear programming. The non-linear programming was mostly about the Kuhn-Tucker conditions, and there the work was mostly about the Kuhn-Tucker necessary conditions. Kuhn and Tucker were long at Princeton. The guy who was the Chair of my Ph.D. oral exam had been a Tucker student. Before I got to that grad program, I had carefully studied W. Rudin, 'Principles of Mathematical Analysis' (a.k.a. Baby Rudin) and W. Fleming, 'Functions of Several Variables'. In my first year of grad school, I also had a severe course in H. Royden, 'Real Analysis', the real part of W. Rudin, 'Real and Complex Analysis', Neveu, Breiman, Chung, etc. So, that background gave tools that helped attack the Kuhn-Tucker conditions. Intuitive view of the Kuhn-Tucker conditions (KTC): You are in a cave with an uneven floor and vertical walls and you want to find the lowest point. If you put down a marble and it starts to roll, then you are not at the lowest point. So, to be at the lowest point, it is necessary that the marble not roll. But K-T wanted more: They wanted to say that necessarily the slope of the floor (calculus gradient) and the slopes of the constraints that define the walls are such that the slopes from the walls block moving along the slope of the floor. Right, the slopes from the walls form a cone that contains the slope from the floor (or its negative depending on maximizing or minimizing and the direction of the constraints, etc.) -- it's all about a cone. Well, this stronger statement is true in nice cases, and for a nice case have to have some assumptions that the constraints are nice. For that there are various KI constraint qualifications, KTCQ, that are enough to make the KT statements about slopes true. There are lots of KT CQs, and one question was, which imply the others? For two famous KT CQs, one due to KT and one due to Zangwill, it was not known if they were independent. So, as a grad student, I settled that -- they are independent. The proof was by counterexample -- I found some bizarre constraints. To know that such bizarre (goofy, pathological, etc.) constraints could exist, I needed essentially a theorem For a positive integer n, the real numbers R, Euclidean n-space R^n with the usual topology, and a subset C of R^n closed in that topology, there exists a function f: R^n --> R such that f is zero on closed set C, strictly positive otherwise, and infinitely differentiable. So, I proved that. For the KT CQ I didn't need all of infinitely differentiable, but I got that also. This result is curious in part because some examples of a close set C can be surprisingly intricate, e.g., the Mandelbrot set, a sample path of Brownian motion, Cantor sets of positive measure, etc. As I went to publish, I discovered that my work also answered a question asked but not answered in a paper by Arrow, Hurwicz, and Uzawa. Of course Arrow got his Nobel Prize. A few years ago, so did Hurwicz. Last I heard, Uzawa was still waiting! Cute: As a grad student I answered a question asked but not answered by Arrow, Hurwicz, and Uzawa. Reading Rudin and Fleming helped! Gee, in the OP, I see that Arrow was also interested in decision making under uncertainty. Well, my dissertation research was in best decision making over time under uncertainty -- stochastic optimal control. I never took a course in economics. My Ph.D. advisor thought that I would need such a course if only later in my career to fend off nonsense objections from economists -- I've never needed that! So, I signed up for an econ course, went the first day, sat in the front row, said nothing, and took careful notes. After the class when just the professor and I were there, I asked him what he was assuming for his supply and demand curves -- continuous, uniformly continuous, differentiable, continuously differentiable, infinitely differentiable, convex, pseudo-convex, quasi-convex, etc.? He said nothing. Soon I got a call from my department secretary to call my Ph.D. advisor -- I was out of the econ course! Still, the OP shows that I was closer to some mathematical economics than I knew! Maybe someday some people in data science or artificial intelligence will exploit the KTC!