A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schr\"odinger operator by decomposing the problem into a stability estimate for H\"older's inequality and a stability estimate for a Gagliardo--Nirenberg--Sobolev inequality. In this note we point out that, if the H\"older step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen--Frank--Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=\gamma+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.