  In a recent paper, E. Carlen and A. Figalli prove a stability estimate - also
known as a quantitative inequality - for a sharp Gagliardo-Nirenberg inequality
and use this result to solve a Keller-Segal Equation. The Gagliardo-Nirenberg
Inequality that Carlen and Figalli prove their stability estimate for is part
of a larger family of sharp Gagliardo-Nirenberg inequalities, for which the
sharp constants and extremals were identified in some cases by Carrillo and
Toscani, and then for the remaining cases by Del Pino and Dolbeault. We prove a
stability estimate for the entire family of sharp Gagliardo-Nirenberg
inequalities for which Del Pino and Dolbeault calculated the sharp constants
and extremals. Establishing stability estimates of inequalities is an active
topic. A key piece of our proof is the application of a continuous dimension
generalization of the Bianchi-Egnell Stability Estimate proven by the author.
