Rolling contact and support-function inequalities for convex curves and surfaces

If we throw a heavy convex polyhedron with arbitrary interior mass distribution onto a horizontal floor then it will come to rest in a stable position on one of its faces. That is there exists for an arbitrary point $P$ lying in the interior of the convex polyhedron one face $F$ (at least) with the following property: The perpendicular dropped from $P$ onto the plane in which $F$ lies has its foot in the interior of the face $F$. Give a purely geometrical proof free from mechanical considerations for the existence of the face $F$.

If a curve viewed from the point $O$ appears everywhere convex, then it subtends a visual angle $< 180°$; if it appears everywhere concave and extends to infinity it subtends a visual angle $> 180°$.

Let $O$ be the origin of a system of polar coordinates $r$, $\varphi$ and $r = 1/f(\varphi )$ the equation of the curve, where $f(x)$ is twice differentiable. In calculating the radius of curvature we are led to the following two theorems:

1. If the function $f(x) > 0$ in the interval $a \leq x \leq a+\pi$, then there exists a point $\xi$, $a < \xi < a+\pi$, such that

   $$f(\xi ) + f''(\xi ) > 0 .$$

2. If $f(x) > 0$, $f(x)+f''(x) > 0$ for $a < x < b$ and $f(a) = f(b) = 0$, then $b-a > \pi$. Prove these theorems by purely analytical methods.

On a closed convex curve with continuous curvature there are always at least three different pairs of points with the following property: The tangents at the two points of a pair are parallel and the curvature of the curve at the two points is the same.

If a convex curve of circumference $l$ and of area $a$ can roll unimpeded in the interior of another convex curve of circumference $L$ and area $A$, then we have

Denote the volume, the surface area and the integral of the mean curvature of two convex surfaces by $v$, $s$, $m$ and $V$, $S$, $M$ respectively. If the first body can roll unimpeded in the interior of the second, then we have the inequalities

Two spheres, one of radius $r$, the other of radius $R$, are externally tangent to each other. We denote by $B$ the smallest convex solid containing both spheres (a tight bag containing two balls). Let $V$ stand for the volume of $B$, $S$ for the area of its surface, and $M$ for its Minkowski constant (surface integral of the mean curvature). Prove that