Does every compact subset of Euclidean space have a centroid? Let $A$ be a compact subset of $\mathbb{R}^n$. Consider the set of hyperplanes in $\mathbb{R}^n$ that separate $A$ into two pieces with equal measure (Lebesgue). Now take the intersection over all such planes. Call a point $x$ a centroid of $A$ if it lies in this intersection. It seems like this definition is equivalent to the usual definitions for centroid, but if not please let me know. My main question is: Is the resulting intersection always non-empty? 
I'm mostly interested in the cases $n=2$ and $n=3$, but it seems like the answer will probably be the same for all $n > 1$.
For example, consider the sphere $S^2 \subset \mathbb{R}^3$. Any plane through the origin will cut $S^2$ into two sets of equal measure, and no other plane will, so there is a unique centroid. 
EDIT: This post may be related, but I don't think it answers my question. https://mathoverflow.net/questions/248206/a-question-about-the-centroids-of-compact-subsets-of-euclidean-spaces
EDIT 2: I realized that I'm asking the wrong question, since the answer to the above is clearly no, as demonstrated in the comments. My revised question is: Given a compact set $A$, does there exists a point $x$ such that any hyperplane through $x$ separates $A$ into two sets of equal measure? In the example of a union of two disjoint disks given below, the centroid of the set would be such a point. Does such a point always exist? 

Not every line through the centroid of a triangle splits the triangle in pieces with the same area:

and not every line splitting the triangle in halves goes through a common point: $GH$ is a "splitting segment" iff $CG\cdot CH=\frac{1}{2}CB\cdot CA$, but here

the red splitting segments are not concurrent since $\frac{1}{\sqrt{2}}\neq\frac{2}{3}$.
