Parallel Sections of a Convex Body by Planes Tangent to an Inner Sphere

Key Points

• The convex body must be a solid sphere if its parallel sections are translates by the tangency vector.
• Any two parallel cross-sections of the body are translates of each other based on their tangency points.
• This geometric theorem highlights a crucial link between convex shapes and inner spheres in geometry.
• Understanding the implications of this theorem can deepen insights into the properties of convex bodies.