On the girth of convex bodies

Abstract.

For a three-dimensional convex body and a given direction the corresponding girth is defined as the perimeter of the orthogonal projection of the body onto a plane orthogonal to the assigned direction. Obvious examples show that there are pairs of convex bodies that have in every direction equal girths but are not translates of each other. Here a modification of the concept of the girth, called semi-girth, is introduced, and it is shown that convex bodies are uniquely determined (up to translations) by their semi-girths. In addition, corresponding stability results are proved and an inequality concerning the central symmetry of convex domains is established.