Let $A$ and $B$ be two compact convex subsets of $\mathbb{R}^n, n\geq 2$. Assume $x_A$ and $x_B$ are their respective centroid. If we form the Minkowski sum $C=A+B = \{x+y\mid x\in A, y\in B\}$, what is the centroid $x_C$ of $C$? How is it related to $x_A+x_B$?
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1$\begingroup$ There should be only one question in one post. $\endgroup$– Iosif PinelisCommented Mar 20 at 12:48
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2$\begingroup$ I don't think MO has "only one question per post rule". Do you define centroid as the center of gravity? In convex geometry one studies functions invariant under Minkowski addition. In particular, if $f$ is a function from the set of convex compact subsets of $\mathbb R^n$ into $\mathbb R^n$ such that $f$ is Minkowski additive, equivariant under rigid motions and continuous in the Hausdorff distance, then $f$ is the Steiner point (which is not always the same as the center of gravity). The result can be found, e.g. in the book [Schneider, Convex Bodies: The Brunn-Minkowski Theory, 2013]. $\endgroup$– Igor BelegradekCommented Mar 20 at 14:00
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$\begingroup$ continued: if you have no access to Schenider's book see p.222 in math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/McMullen/…. $\endgroup$– Igor BelegradekCommented Mar 20 at 14:03
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$\begingroup$ @IgorBelegradek : (i) Please look at this guideline: "avoid trying to answer questions which [...] request answers to multiple questions". (ii) The centroid (also known as center of mass, center of gravity, or barycenter) is a standard notion. Even though there are other points that can be assigned to a convex set, such as the Steiner point, they are definitely not the centroid. $\endgroup$– Iosif PinelisCommented Mar 20 at 14:29
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$\begingroup$ Do you have a response to the answer below? $\endgroup$– Iosif PinelisCommented Mar 22 at 19:26
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1 Answer
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The tautological answer to the question "what is the centroid $x_C$ of $C$?" is "the centroid $x_C$ of $C$ is the centroid $x_C$ of $C$". It is hardly possible to give a better answer to this question without having the terms in which to express $x_C$ specified.
The answer to your second question, "How is it related to $x_A+x_B$?" is "in no reasonable way".
Indeed, for any natural $n\ge2$ and any points $a,b,c$ in $\Bbb R^n$ it is easy to find two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_{A+B}=x_C=c$.
To do that, start with $n=2$, let $A$ be the convex hull of the set $\{(-1,0),(1,0)\}$, and let $B$ be the convex hull of the set $\{(-2,0),(1,-1),(1,1)\}$. Then $x_A=x_B=(0,0)$ and $C=A+B$ is the convex hull of the set $\{(2,1),(2,-1),(-3,0),(0,1),(0,-1)\}$, so that $x_C=(1/7,0)\ne(0,0)=x_A+x_B$.
Now -- by rescaling, embedding $\Bbb R^2$ into $\Bbb R^n$, rotating, and shifting -- for any natural $n\ge2$ and any points $a,b,c$ in $\Bbb R^n$ such that $c\ne a+b$ we get two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_C=c$.
It is even easier, for any natural $n\ge2$ and any points $a,b$ in $\Bbb R^n$
to get two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_C=a+b$.
answered Mar 20 at 12:48
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$\begingroup$ @IgorBelegradek : (i) I don't know what you mean by nitpicking here. (ii) The centroid (also known as center of mass, center of gravity, or barycenter) is a standard notion. Even though "there are other centers that can be assigned to a convex set, e.g. circumcenter, Steiner point", they are definitely not the centroid. $\endgroup$ Commented Mar 20 at 14:26
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$\begingroup$ @IgorBelegradek : You are ascribing thoughts or even "knowledge" to me that I never had, and then based on that accusing me of bad faith, while nonetheless writing that the question is "sloppy". I do see here two questions. One is of "what is" type, with completely unspecified (and not reasonably understood from the context) terms in which to describe the object. The second, "How is it related" question is a bit more specific, and it has been completely answered. So, can you reconsider your "because I think you know"? $\endgroup$ Commented Mar 20 at 14:52
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$\begingroup$ @IgorBelegradek : That there is a book discussing "curvature centroids" and "area centroids" does not make the term "centroid" less standard or requiring any clarification. Rather, I think this just muddies the waters. $\endgroup$ Commented Mar 20 at 14:56
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$\begingroup$ @IgorBelegradek : I am sorry to see that you have just removed your comments on this answer, without responding substantially to my points. $\endgroup$ Commented Mar 20 at 15:11