When Does a Shape Know It's Almost a Sphere?

A recurring theme in geometry is that equality in a sharp inequality forces a shape to be perfectly round. But what happens near equality? If a domain nearly optimises the isoperimetric ratio, or the quermassintegral deficit, does it have to nearly look like a ball? I will discuss recent quantitative stability results for quermassintegral inequalities in hyperbolic space and Minkowski-type inequalities in warped product spaces, explaining how geometric flows give a natural route to such estimates and what new difficulties arise in curved settings.