In　this　paper,　it　is　proved　that　for　one-hidden-layer　ReLU　networks　all　differentiable　local　minima　are　global　inside　each　differentiable　region.　Necessary　and　sufficient　conditions　for　the　existences　of　differentiable　local　minima,　saddle　points　and　non-differentiable　local　minima　are　given,　as　well　as　their　locations　if　they　do　exist.　Building　upon　the　theory,　a　linear　programming　based　algorithm　is　designed　to　judge　the　existence　of　differentiable　local　minima,　and　is　used　to　predict　whether　spurious　local　minima　exist　for　the　MNIST　and　CIFAR-10　datasets.　Experimental　results　show　that　there　are　no　spurious　local　minima　for　most　typical　weight　vectors.　These　theoretical　predictions　are　verified　by　demonstrating　the　consistency　between　them　and　the　results　of　gradient　descent　search.　?　2021　Elsevier　B.V.　All　rights　reserved.　Superscript/Subscript　Available＜/comment