Maximizing and Minimizing the Width

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

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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?
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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.