We define new natural variants of the notions of weighted covering andseparation numbers and discuss them in detail. We prove a strong dualityrelation between weighted covering and separation numbers and prove a fewrelations between the classical and weighted covering numbers, some of whichhold true without convexity assumptions and for general metric spaces. As aconsequence, together with some volume bounds that we discuss, we provide abound for the famous Levi-Hadwiger problem concerning covering a convex body byhomothetic slightly smaller copies of itself, in the case of centrallysymmetric convex bodies, which is qualitatively the same as the best currentlyknown bound. We also introduce the weighted notion of the Levi-Hadwigercovering problem, and settle the centrally-symmetric case, thus also confirmNasz\'{o}di's equivalent fractional illumination conjecture in the case ofcentrally symmetric convex bodies (including the characterization of theequality case, which was unknown so far).
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