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We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwigers conjecture, and inequalities about M-positions for geometric log-concave functions. In particular, we obtain stro
This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical norm. This
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenber
A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of
Let $0<p,qleq infty$ and denote by $mathcal{S}_p^N$ and $mathcal{S}_q^N$ the corresponding Schatten classes of real $Ntimes N$ matrices. We study the Gelfand numbers of natural identities $mathcal{S}_p^Nhookrightarrow mathcal{S}_q^N$ between Schatten