  This paper shows a mathematical formalization, algorithms and computation
software of volume optimal cycles, which are useful to understand geometric
features shown in a persistence diagram. Volume optimal cycles give us concrete
and optimal homologous structures, such as rings or cavities, on a given data.
The key idea is the optimality on $(q + 1)$-chain complex for a $q$th homology
generator. This optimality formalization is suitable for persistent homology.
We can solve the optimization problem using linear programming. For an alpha
filtration on $\mathbb{R}^n$, volume optimal cycles on an $(n-1)$-th
persistence diagram is more efficiently computable using merge-tree algorithm.
The merge-tree algorithm also gives us a tree structure on the diagram and the
structure has richer information. The key mathematical idea is Alexander
duality.
