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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
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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 result is convenient to apply in real applications and could yield refined local Rademacher complexity bounds for function classes satisfying general entropy conditions. We demonstrate the power of our complexity bounds by applying them to derive effective generalization error bounds.
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 and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Nasz{o}dis equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).
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-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain sever
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 path-matchings. Given positive integers $r, p_1, dots, p_r$, define $R^{PM}(p_1, dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $iin [r]$. Our main result is that for $rgeq 2$ and $p_1geq dotsgeq p_rgeq 2$, if $p_1geq 2r-2$, then [R^{PM}(p_1, dots, p_r)= p_1- (r-1) + sum_{i=2}^{r}leftlceilfrac{p_i}{3}rightrceil.] Perhaps surprisingly, we show that when $p_1<2r-2$, it is possible that $R^{PM}(p_1, dots, p_r)$ is larger than $p_1- (r-1) + sum_{i=2}^{r}leftlceilfrac{p_i}{3}rightrceil$, but in any case we determine the correct value to within a constant (depending on $r$); i.e. [p_1- (r-1) + sum_{i=2}^{r}leftlceilfrac{p_i}{3}rightrceil leq R^{PM}(p_1, dots, p_r)leq leftlceil p_1-frac{r}{3}+sum_{i=2}^rfrac{p_i}{3}rightrceil.] As a corollary we get that in every $r$-coloring of $K_n$ there is a monochromatic path-matching of order at least $3leftlfloorfrac{n}{r+2}rightrfloor$, which is essentially best possible. We also determine $R^{PM}(p_1, dots, p_r)$ in all cases when the number of colors is at most 4. The proof of the main result uses a minimax theorem for path-matchings derived from a result of Las Vergnas (extending Tuttes 1-factor theorem) to show that the value of $R^{PM}(p_1, dots, p_r)$ depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case.
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 classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Chavez-Dominguez and D. Kutzarova.
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