We have studied about volume of solid of revolution of plane regions in undergraduate Calculus classes. we might have observed that volume of solid of revolution varies whenever we allow to vary the line of revolution. It raised questions! like, Does there exist a point in every simply connected compact plane region such that the volume of solid of revolution about any line passing through the point is invariant? what about, when said region is Convex also, and what about higher dimension?

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A simple example is center of a circular region is a Point of Volume Invariance.
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"Suppose $X$ is a convex, compact simply-connected region in $\mathbb R^2$, and $ℓ$ a line in $\mathbb R^2$. Let $X^l_p$ denotes the set of all lines in $\mathbb R^2$ through a fixed point $p$ in $X$ and $V_l$ denotes the volume of solid of revolution of $X$ about the line $l$. Does there exist a point $p$ in $X$ such that the set $\{V_l| l \in X_p^l\}$ is a singleton? What will happen if $X$ is not convex? what about higher dimensions i. e. if $X$ is in $\mathbb R^n$?

titleof the question!) You could write something like "Suppose $X$ is a compact simply-connected region in $\mathbb R^2$, and $\ell$ a line in $\mathbb R^2$, and let $\hat X_\ell$ denote..." Such edits aren't strictly necessary, but I'd find the question easier to read. Do check out mathoverflow.net/howtoask . Also, you may want to provide more motivation? $\endgroup$ – Theo Johnson-Freyd Oct 29 '11 at 14:31