John functions, quadratic integral forms and o-minimal structures

JOHN FUNCTIONS FOR o-MINIMAL DOMAINS

A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds for o-minimal domains. We prove their definability and give explicit estimates for number of John functions required for a given set to obtain uniform positi...

Representation of Quadratic Forms by Integral Quadratic Forms

Near Integral Points of Sets Definable in O Minimal Structures

Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly exponentially near to many integral points, then there is an unbounded set, which is R definable without parameters, and which is exponentially near to S.

Quadratic Functions, Optimization, and Quadratic Forms

Quadratic Forms and Height Functions