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 $\mathbb R^n$? 



Existence of point of Volume invariance.
[Note: This is an answer to the original version of the question, without the convexity condition.]


No. Consider two disks of small radius $\epsilon$, joined by a very thin ($<< \epsilon$) filament. The volume of the solid of revolution will be approximately proportional to the sum of lengths of the circles of revolution corresponding to the centers of the disks. The length of one of these circles has the form $|r \operatorname{sin}(\theta + c)|$, where $\theta$ is the angle of the line we are revolving around and $r$ and $c$ are constants. (I'm assuming we have already fixed the point through which all these lines pass.) It is not possible to add two functions of this form and get a constant function (of $\theta$), unless $r=0$ in both cases. But this will not happen if the centers of the disks are distinct. (This calculation ignores the possibility that the two circles of revolution coincide, but this will happen for at most two values of $\theta$.)


