Poincaré Inequality

The Poincaré Inequality is a mathematical concept that relates the average behavior of a function to its variation. In simple terms, it states that if you have a smooth function defined over a specific domain, the "spread" or "difference" of that function from its average value can be bounded by how much the function varies within that domain. This principle is often used in analysis and geometry, providing a way to connect the values of functions to their derivatives, helping to understand how shapes and spaces behave in mathematics and physics.