Functional Analysis
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Paper Metric Markov cotype of \ell_{1} and Hilbert-space Lipschitz extension
Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a…
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Paper Reflexive midpoint convexity and diamond distortion
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same…
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Paper Lipschitz equivalent Banach spaces need not be linearly isomorphic
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up…
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Paper The complete Crouzeix conjecture
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has \lVert P[A]\rVert\le2\sup{z\in W(A)}\lVert P(z)\rVert, where…
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Paper A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively.…
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Paper Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without…
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Paper A counterexample to metric-entropy duality
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and…
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Paper The cotype–cotype conjecture under the approximation property
Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different…
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Paper Markov type characterizes superreflexivity
Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes…
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Paper Tingley’s sphere-isometry problem
Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is…
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Paper Independence of the separable quotient problem
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable…
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