# Convex misfit of finite-dimensional convex bodies

The finite-dimensional convex bodies topic belongs to the combinatorial geometry. The direction below has a bit more of an algebraic flavor. Let me/us know if some or all of the notions below are already known (and I apologize for my ignorance).

Let the input convex bodies in $\ A\subseteq \mathbb R^n\$ have Lebesgue measure $1,\ |A|=1.\$ For two of them, $\ A\$ and $\ B,\$ let $\ A+B\$ be the Minkowski sum of them. Then

$$m(A\ B)\ =\ \frac{|A+B|}{2^n} - 1$$

is called the misfit of $\ A\$ and $\ B.\$ It's only natural that the misfit $\ m(A\ A)\$ of a body with itself is zero.

Next, let $\ \mathcal G\$ (a caligraphic $G$) be a group of isometries of $\ \mathbb R^n\$ (it'll be enough to consider subgroups of linear isometries $\ O(n)).\$ Then the misfit over $\ \mathcal G\$ is defined by

$$m_\mathcal G(A\ B)\ :=\ \inf_{G\in\mathcal G} m(G(A)\ B)$$

Obviously, $m_\mathcal G(A\ B)\ =\ m_\mathcal G(B\ A),\$ and it's well-known that the $\inf$ is reached by a certain $\ G\in\mathcal G.$

Finally, let $\ A'\ :=\ \{(-x_1\ x_2\ \ldots x_n)\ :\ (x_1\ \ldots\ x_n)\in A\}$ be a mirror reflection of $\ A.$

QUESTIONS:

1. Compute (or estimate) the worst misfit $\quad M_n\ :=\ sup_{_{A\ B\subseteq \mathbb R^n}}\ m_{O(n)}(A\ B);$

2. Find $\ A\ B\subseteq \mathbb R^n\$ such that $\ m_{O(n)}(A\ B)\ =\ M_n;$

3. Compute (or estimate) the worst oriented misfit $\quad S_n\ :=\ sup_{_{A\ B\subseteq \mathbb R^n}}\ m_{SO(n)}(A\ B);$

4. Find $\ A\ B\subseteq \mathbb R^n\$ such that $\ m_{SO(n)}(A\ B)\ =\ S_n;$

5. Compute (or estimate) the worst mirror misfit $\quad Q_n\ :=\ sup_{_{A\subseteq \mathbb R^n}}\ m_{SO(n)}(A\ A');$

6. Find the worst mirror misfit $\ A\subseteq \mathbb R^n\$ such that $\ m_{SO(n)}(A\ A')\ =\ Q_n;$

7. Is the right-angled half of the equilateral triangle, of area $1,\$ an example of the worst mirror misfit in $\ R^2\, ?$

It'd be nice to compare the known misfit related searches, e.g. by @Wlodek Kuperberg, with the present note.

EDIT

Ivan has destroyed my questions $1-4$. Questions $5-7$ still seem to make sense.

I'll provide modified questions $1-7$ in a different note (it'd be confusing to mix them here).

• I would suggest to consider the group of volume-preserving affinities, then call the notion affine misfit. – Wlodek Kuperberg Jun 7 '18 at 22:44
• @WlodekKuperberg, thank you, you're reading in my mind. :-) I still prefer convex misfit to make the term more focused (especially that the misfit still applies to different groups; e.g. the mirror misfit sounds good for isometries). – Wlod AA Jun 7 '18 at 22:53

The misfit and the oriented misfit can be arbitrarily large. Let $A$ be a ball, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. We have $|G(A)+B| = |A+B|$. The volume of $A+B$ tends to infinity with the length of the cylinder (because $A+B$ contains many copies of $A$).