# What does it essentially mean if the neural network has convex error surface?

Suppose if I am building a Linear Regression model with one fully connected layer and a sigmoid with minimizing mean squared error as objective. Why would the error surface be convex?

Does finding the optimal parameters for this network mean we cannot do better than this? Under what assumptions, this solution would be optimal? If we relax the linearity assumption and add some non-linearity to the network can we do better than this? Why so?