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The question is in the title. My intuition would tell me that it should either be called an $n$-hemisphere, a hemi-$n$-sphere, or a semi $n$-sphere. However, I am primarily curious if there has been any established terminology for the object already. Ultimately, I'd like to know if anyone has a source or article that refers to this object one way or another.

David G.
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    You are not required to use Latin or Greek: a "half $n$-sphere" will do, as in https://math.stackexchange.com/questions/4749321/integral-for-the-surface-area-of-a-half-n-sphere or https://math.stackexchange.com/questions/4724631/integral-of-differential-n-form-over-half-n-sphere or https://math.stackexchange.com/questions/4532363/how-do-i-scale-a-probability-density-defined-on-an-n-ball-when-mapping-it-to-a-h – Henry Jul 03 '24 at 23:10

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