@guest61: the estimate you want holds in the other sense, as indicated in my answer.

@Héhéhé: If we're talking about Dirichlet eigenvalues the tangential derivative of the eigenfunction is zero, since it is constant in the tangential direction. The norm of the gradient or the normal derivative gives the same thing.

@GeoffreyIrving: This is not a counter example in dimension 3. The minimal projection will be quite small (a rectangle with one side very small), while the maximal section will be at least equal to the area of the equilateral triangle (areas of parallel sections with respect to a direction give a concave function).

In 2D the minimal projection is the minimal width and the maximal section is the diameter, i.e. the maximal width. Therefore the inequality is obvious.

Is the volume of the intersection of $n$ unit balls centered at the vertices of the $n$-dimensional unit simplex computed somewhere? This is a set of unit diameter so it contains a set of unit constant width.

Not knowing your background and sub-field, it is hard to give advice/encouragements. When I finished my PhD I had quite a few work ideas which were solvable given enough time. With more time, you'll find that there are so many subjects to work on that you need to choose. For me it's not 'fighting'. It's a constant learning experience. Learn, experiment, sometimes solve, then repeat. There are important/difficult questions: keep them in mind until you gather enough tools to tackle some of them. In the meantime, always have some 'feasible' problems at hand to work on.

I'm not sure why you would add the $r(t)$ term in the above definition. Maybe you could imagine some weird counterexamples, where differentiation in this sense is more general than the more simple directional derivative. The Wiki definition of Hadamard derivative does not include the higher order remainder $r$.

If the 2D question is not yet solved, I doubt there's a quicker idea in 3D...

Concerning the second question, do you have a conjecture? The 2D analogue is solved?

For a polyhedron you could easily imagine a discrete algorithm. Take an orientation of the polyhedron and move it vertically such that the plane $z=0$ bisects your polyhedron. You should be able to find algorithms computing all geometric quantities for the bisecting sections. These algorithms should be quick, since only geometric quantities are involved. Since differentiating these objective functions might be tricky, you could use gradient free optimization (Golden Search in 1D, Nelder in 3D) for the optimization part.

@MateuszKwaśnicki: I don't think the Dirichlet Laplace eigenvalues need to be considered here. The volumic source term is zero. You need rather the Steklov eigenvalues: $-\Delta u=0$, $\partial_n u = \sigma_k u$ on the boundary?