The ebook describes how sensible inequalities are frequently manifestations of traditional mathematical buildings and actual phenomena, and the way a number of normal ideas validate huge sessions of analytic/geometric inequalities, previous and new. This perspective ends up in "systematic" techniques for proving the main easy inequalities, but additionally for bettering them, and for devising new ones--sometimes at will and sometimes on call for. those basic rules additionally provide novel methods for estimating top constants and for identifying even if those are attained in acceptable functionality areas. As such, advancements of Hardy and Hardy-Rellich style inequalities related to radially symmetric weights are variational manifestations of Sturm's concept at the oscillatory habit of sure usual differential equations. however, so much geometric inequalities, together with these of Sobolev and Log-Sobolev variety, are easily expressions of the convexity of yes unfastened strength functionals alongside the geodesics at the Wasserstein manifold of chance measures outfitted with the optimum mass delivery metric. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev variety inequalities are then received through interpolating the above periods of inequalities through the classical ones of Hölder. the delicate Moser-Onofri-Aubin inequalities at the two-dimensional sphere are hooked up to Liouville kind theorems for planar suggest box equations.

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