  This paper is concerned with necessary and sufficient second-order conditions
for finite-dimensional and infinite-dimensional constrained optimization
problems. Using a suitably defined directional curvature functional for the
admissible set, we derive no-gap second-order optimality conditions in an
abstract functional analytic setting. Our theory not only covers those cases
where the classical assumptions of polyhedricity or second-order regularity are
satisfied but also allows to study problems in the absence of these
requirements. As a tangible example, we consider no-gap second-order conditions
for bang-bang optimal control problems.
