I don't know how to go about such questions. It's not exactly my area, so maybe it is stupid, but curiosity is winning.

So I have a piece of bread $P$ of a really non-regular shape (let's make it convex though), and I want to make a sandwich from it, i.e. two pieces of bread with some stuff between them. Obviously I have to cut my piece into two pieces. My knife is straight and I can only make one cut, otherwise the cooking will be too messy. In other words the two pieces that I obtain are the intersections $P^l_1$ and $P^l_2$ of $P$ with half-planes with respect to a certain line $l$.

Then, in order to make a sandwich I have to place $P_1$ above $P_2$ and put stuff in between. The most effective use of bread happens when these pieces are approximately aligned, i.e. only most $P_2$ is covered with $P_1$ and another way around. How to make it that way?

Let's state the questions in the mathematical language.

Let $P$ be a convex body in $\mathbb{R}^{n}$, let $l$ be a hyperplane and let $P^l_1$ and $P^l_2$ be the pieces in which $l$ cuts $P$. Let $\lambda_n$ be the usual volume in $\mathbb{R}^{n}$.

The quantity we are interested in is $$\sup\{ \frac{\lambda_n(P^l_1\bigcap f(P^l_2))}{\lambda_n(P)},f\mbox{ - isometry of }\mathbb{R}^{n},~l\mbox{ - hyperplane}\}.$$

Q1: Is it bounded from below by some absolute constant, perhaps depending on $n$?

Q2: Is there an algorithm to construct an optimal, or almost optimal $l$ and $f$?

I don't know how to go about such questions. It's not exactly my area, so maybe it is stupid, but curiosity is winning.

So I have a piece of bread $P$ of a really non-regular shape (let's make it convex though), and I want to make a sandwich from it, i.e. two pieces of bread with some stuff between them. Obviously I have to cut my piece into two pieces. My knife is straight and I can only make one cut, otherwise the cooking will be too messy. In other words the two pieces that I obtain are the intersections $P^l_1$ and $P^l_2$ of $P$ with half-planes with respect to a certain line $l$.

Then, in order to make a sandwich I have to place $P_1$ above $P_2$ and put stuff in between. The most effective use of bread happens when these pieces are approximately aligned, i.e. only most $P_2$ is covered with $P_1$ and another way around. How to make it that way?

Let's state the questions in the mathematical language.

Let $P$ be a convex body in $\mathbb{R}^{n}$, let $l$ be a hyperplane and let $P^l_1$ and $P^l_2$ be the pieces in which $l$ cuts $P$. Let $\lambda_n$ be the usual volume in $\mathbb{R}^{n}$.

The quantity we are interested in is $$\sup\{ \frac{\lambda_n(P^l_1\bigcap f(P^l_2))}{\lambda_n(P)},f\mbox{ - isometry of }\mathbb{R}^{n},~l\mbox{ - hyperplane}\}.$$

Q1: Is it bounded from below by some absolute constant? EDIT: As suggested by UriBader John's inscribed ellipsoid provides with an estimate from below with an absolute constant that depends on $n$. Thus, I am leaving the stronger version of the question, i.e. if there is an absolute constant that does not depend on the dimension.

Q2: Is there an algorithm to construct an optimal, or almost optimal $l$ and $f$?