  Given an arbitrary quantum state ($\sigma$), we obtain an explicit
construction of a state $\rho^*_\varepsilon(\sigma)$ (resp.
$\rho_{*,\varepsilon}(\sigma)$) which has the maximum (resp. minimum) entropy
among all states which lie in a specified neighbourhood ($\varepsilon$-ball) of
$\sigma$. Computing the entropy of these states leads to a local strengthening
of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes
inequality. Our bound is local in the sense that it depends on the spectrum of
$\sigma$. The states $\rho^*_\varepsilon(\sigma)$ and
$\rho_{*,\varepsilon}(\sigma)$ depend only on the geometry of the
$\varepsilon$-ball and are in fact optimizers for a larger class of entropies.
These include the R\'enyi entropy and the min- and max- entropies. This allows
us to obtain local continuity bounds for these quantities as well. In obtaining
this bound, we first derive a more general result which may be of independent
interest, namely a necessary and sufficient condition under which a state
maximizes a concave and G\^ateaux-differentiable function in an
$\varepsilon$-ball around a given state $\sigma$. Examples of such a function
include the von Neumann entropy, and the conditional entropy of bipartite
states. Our proofs employ tools from the theory of convex optimization under
non-differentiable constraints, in particular Fermat's Rule, and majorization
theory.
