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This paper shows a mathematical formalization, algorithms and computationsoftware of volume optimal cycles, which are useful to understand geometricfeatures shown in a persistence diagram. Volume optimal cycles give us concreteand 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 homologygenerator. This optimality formalization is suitable for persistent homology.We can solve the optimization problem using linear programming. For an alphafiltration on $\mathbb{R}^n$, volume optimal cycles on an $(n-1)$-thpersistence diagram is more efficiently computable using merge-tree algorithm.The merge-tree algorithm also gives us a tree structure on the diagram and thestructure has richer information. The key mathematical idea is Alexanderduality.

Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology