Problems and Theorems in Analysis · 9.1

Let $k_1, k_2, \ldots , k_6$ be the lengths of the six edges of a tetrahedron and $\alpha_1, \alpha_2, \ldots , \alpha_6$ the radian measures of the angles formed by the faces of the tetrahedron that meet at the respective edges. Then we have

The given bounds are the best possible.

Consider a triangle. Let $a$, $b$ and $c$ denote its sides and $A$ its area. Prove that

with equality only for equilateral triangles.

(continued). If $p > 0$

with equality only for equilateral triangles.

(continued). Let $r$ and $R$ denote the radii of the inscribed and the circumscribed circle of a triangle, respectively. Prove that

with equality only for equilateral triangles.

Formulate a proposition that has an analogous relation to 17.1 as 17 has to 16.

(continued). Try to find simple particular cases for which the proposition formulated above can be conveniently tested.