Balls have the worst best Sobolev inequalities

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Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems (2008, unknown): “[1], [15], [16], [28].”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “On the pure side, these include connections to inequalities [92] [131] [94] [32] [90] [52], geometry (including sectional [85] [73], Ricci [88] [87] [127] [99] and mean [75] curvature), nonlinear partial differential equations [14] [18] [16] [135] [89], and dynamical systems (weak KAM theory [9]; no”
The sharp Sobolev inequality in quantitative form (2009, background): “key ingredient here is a trace inequality from [ MV ], Theorem 1.3, which, in the one-dimensional case, tells us that µ n!n Z r”
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences (2013, methodology): “In recent years, mass transportation method has been used successfully to prove some sharp inequalities in Analysis and Geometry, for example, see [3, 5, 11, 12, 16, 18, 19, 22, 24].”
Best constant in Sobolev trace inequalities on the half-space (2006, methodology): “We begin by establishing with the techniques of [11, 20 ] an inequality valid on two arbitrary smooth and compactly supported probability densities F and G. Then, in a second part, we apply the inequality to F = f”