One of the holy grails of algorithm design is finding a strongly polynomial algorithm for linear programming, i.e., an algorithm whose runtime is bounded by a polynomial in the number of variables and constraints and is independent of the size of the representation of the parameters (assuming unit cost arithmetic). Would resolving this question have implications outside of better algorithms for linear programming? For instance, would the existence/non-existence of such an algorithm have any consequences for geometry or complexity theory?
Edit: Maybe I should clarify what I mean by consequences. I'm looking for mathematical consequences or conditional results, implications that are known to be true now. For instance: "a polynomial algorithm for LP in the BSS model would separate/collapse algebraic complexity classes FOO and BAR", or "if there is no strongly polynomial algorithm then it resolves such-and-such conjecture about polytopes", or "a strongly polynomial algorithm for problem X which can be formulated as an LP would have interesting consequence blah". The Hirsch conjecture would be a good example, except that it only applies if simplex is polynomial.