Limiting behaviour of the generalized simplex gradient as the number of points tends to infinity on a fixed shape in R^n

This work investigates the asymptotic behaviour of the gradient approximation method called the generalized simplex gradient (GSG). This method has an error bound that at first glance seems to tend to infinity as the number of sample points increases, but with some careful construction, we show that this is not the case. For functions in finite dimensions, we present two new error bounds ad infinitum depending on the position of the reference point. The error bounds are not a function of the number of sample points and thus remain finite.