Maximizing and Minimizing the Width

This post continues from this post

The two questions were given by Prof. Sariel Har-Peled (thanks!) and the answers thereof are due to Prof. Roman Karasev (Thanks!).

-------------
1. What is the minimum width convex shape in the plane, if the perimeter and area are specified?
2. What if the diameter is also specified?
----------
Answers:

For the first question, from physical considerations (forces and tension) the optimal bodies must be like in the attached picture.

To understand this mathematically, one may use the Blaschke compactness for convex bodies and conclude that the optimum is attained. Then investigate the optimal body in the strip from the definition of its minimal width. By local modifications in attempt to decrease the width one may conclude that the boundary must be either on the boundary of the strip or be a circle arc. Moreover, the arc must be tangent to the boundary of the strip at the point they meet, otherwise the widht could be again improved by local modifications.

The second question is not that physical and, I guess, may have multiple solutions. For example, one may intersect bodies of constant width (equal to the given diameter) with the strip of given minimal width. In some range of areas and perimeters this may work. In some may not.

Non-congruent tiling - 14

Here is the previous part of this series:
Revisiting a question posed in the first instalment of this series, back in 2014 (https://nandacumar.blogspot.com/2014/12/filling-plane-with-non-congruent-pieces.html)

In https://arxiv.org/pdf/1905.08144.pdf ?Frettloh and Richter prove: "There is a vertex-to-vertex tiling of the plane by pairwise incongruent triangles of unit area and uniformly bounded perimeter. "

Remark: Their construction seems to achieve a tiling with incongruent equal area triangles whose perimeters lie within an arbitrarily small range (equal is not possible).

Question: Will a similar construction achieve a tiling by pairwise incongruent triangles of unit perimeter and bounded area (area bounded both above and below) with/without the v-to-v property?

Let me also add another recent query posted on mathoverflow: https://mathoverflow.net/questions/421333/on-rigid-packings-of-the-plane-with-a-constraint