"on many dimensions" is a correct and usable phrase in written English and can be used to refer to a situation or topic that involves multiple considerations or facets.
For example, "The impact of climate change is seen on many dimensions, including economic, environmental, and social."
Religiously-based media are growing more expressive in their criticisms, as they take on many dimensions of football culture: violence, felonies, etc.
The collaboration between Malcolm X and Haley took on many dimensions; editing, revising and composing the Autobiography was a power struggle between two men with sometimes competing ideas of the final shape for the book.
That's why the otherness of our distant colleagues is reinforced daily and on many dimensions.
Better-educated individuals tend to do better in life on many dimensions.
Linguistic research sheds light on many dimensions of the human experience.
However, their perceptions on many dimensions shortly after meeting in person are significantly associated with relationship duration.
"This is one of many dimensions on which we'll have to adjust to a reduction in American ability to dictate terms of core interests of ours," said Yochai Benkler, co-director of the Berkman Center for Internet and Society at Harvard.
Few startups fail as a result of spending too much money on servers or from failing to scale to meet customer demand — in fact, optimizing for either of these things is a form of premature scaling, and premature scaling on one or many dimensions (hiring, marketing, sales, product features, and even hierarchy/titles) is the primary cause of death for the vast majority of startups.
Overall, we believe that our indicator improves on the existing ones in many dimensions, but we do acknowledge that it is also subject to a number of important limitations.
With reference to the conceptual framework suggested by this multidimensional model, it is clear that the level of sustainability achieved by a project, a plan or an intervention, depends on how many dimensions are considered in it and on the degree of their mutual relationships [54, 61, 62, 65].
This summary only begins to touch on the many dimensions of the story.
