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The work involves the installation of steel bracing to strengthen the building.
The other courts will remain open during the work, but centre users are likely to notice some extra noise and activity, the Dunedin City Council said in a statement.
The latest work has been timed to coincide with the quietest time of the year for arena bookings. Strengthening of the main court area and reception finished in June.
All work is expected to be completed by early March, at which time the building's earthquake-prone classification can be raised. The expected cost of the entire project is about $2million.