Hersch's inequality theorem

Hersch's inequality provides a mathematical relationship between two important properties of a shape called a convex surface: its *inradius* (the radius of the largest sphere that fits entirely inside it) and its *diameter* (the longest distance across the surface). The theorem states that, for convex shapes on a sphere, the inradius cannot be too large compared to the diameter. Specifically, it sets an upper limit on the inradius relative to the shape's diameter, ensuring that no convex surface on a sphere can have an inradius exceeding a certain fraction of its maximum possible size, maintaining a bound on how "thick" the inner core can be relative to its overall size.