We calculate the induced elastic interaction between
pointwise membrane inclusions that locally interact up to
quadratic order with the membrane curvature tensor. For
isotropic inclusions, we recover the usual interaction
proportional to the inverse fourth power of the separation,
however with a prefactor showing a non-trivial dependence on the
rigidity Γ of the quadratic potential. In the large-Γ limit,
corresponding to "hard" inclusions, we recover the standard
prefactor first obtained by Goulian et
al. (Europhys. Lett. 22, 145 (1993)). In the small-Γ limit,
corresponding to "soft" inclusions, we recover the recent result
of Marchenko and Misbah (Eur. Phys. J. E 8, 477 (2002)). This shows that the
latter result bears no fundamental discrepancy with previous
works, but simply corresponds to the limit of soft inclusions.
We discuss how the same inclusion can be depicted as hard or
soft according to the degree of coarse-graining of the pointwise
description. Finally, we argue that conical transmembrane
proteins should be fundamentally considered as hard inclusions.