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Can a Dimension Feature Class be converted to a shapefile or coverage?
The Dimension feature is considered a 'smart' feature; they act within the parameters of a predefined style and are able to determine their own symbology and how the dimension features themselves should be drawn.
The data formats of coverage and shapefile will not support this functionality.
Instead, the dimension features will appear as polygons in the output format.
See ArcGIS Online Help for more information on 'How to create and manage dimensions'.