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We provide a comprehensive analytic study (rigorous and qualitative) of theconditions for the existence of a a positive kinematic back reaction term$\QQ>0$, in the context of Buchert's scalar averaging formalism applied tospherically symmetric Lema\^itre-Tolman-Bondi (LTB) dust solutions in whichaveraging domains are given as spherical comoving regions containing a symmetrycenter. We introduce proper volume and quasi-local average functionals andfunctions in order to examine the conditions for $\QQ\geq 0$, and in theprocess we also explore the relation between back reaction, spatial curvatureand binding energy for a wide variety of LTB configurations. The back reactionterm is positive for all "hyperbolic" regular domains with negative spatialcurvature, either in the full radial range or in the radial asymptotic range.This result is also valid if these domains contain an inner "elliptic" regionwith positive curvature undergoing local collapse. For some cases in whichpositive spatial curvature decreases asymptotically, the conditions for apositive back reaction can still be met but seem to be more restrictive. Since$\QQ>0$ is a necessary condition for a positive "effective" acceleration thatwould mimic the effect of dark energy (in the context of Buchert's formalism),we examine this issue in LTB models in a follow up paper (part II).

On spatial volume averaging in Lema\^{\i}tre--Tolman--Bondi dust models. Part I: back reaction, spacial curvature and binding energy