We consider the geometry of the trace-ball of quantum states, find maximal and minimal states in a particular partial order called majorization, and use these states to construct local continuity bounds for quantum entropies. We also apply the theory of convex optimization to motivate the construction of the maximal state, and to find general optimality conditions for a particular class of functions subject to a trace-ball constraint. Abstract: “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 Rényi 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 Gâteaux-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."