The authors generalize Cartan’s Lemma, a characterization of real space form, for any four orthonormal vector fields (instead of three) for an indefinite
K
-space.
Let be a monic polynomial of degree , and , . A classic lemma of Cartan asserts that the lemniscate can be covered by balls , whose diameters satisfy For , this shows that has an area at most . Pólya ...showed in this case that the sharp estimate is . We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when is normalized to have norm 1 on some circle, and Remez' inequality.
Let P(z) be a monic polynomial of degree n, and α, ε>0. A classic lemma of Cartan asserts that the lemniscate E(P;ε):={z:|P(z)|≤εn} can be covered by balls Bj,1≤j≤n, whose ...diameters d(Bj) satisfy ∑j=1p(d(Bj))α≤e(4ε)α. For a=2, this shows that E(P;ε) has an area at most Àe(2ε)2. Pólya showed in this case that the sharp estimate is Àε2. We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when P is normalized to have Lp norm 1 on some circle, and Remez’ inequality.
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