Geometric inequalities

Let $A$ be a finite subset of the unit disc in the plane, and let $N(A,r)$ be the set of points at distance $\leq r$ from $A$, where $0<r<1$. Show that the length of the boundary $N(A,r)$ is, at most, $C/r$ for some constant $C$ independent of $A$.

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

Let $a$, $b$, $c$ be the sides, and $\alpha$, $\beta$, $\gamma$ the angles of a triangle in radian measure, $a$ lying opposite to $\alpha$, $b$ to $\beta$ and $c$ to $\gamma$. Then we have

Equality holds in the first inequality if and only if the triangle is equilateral, and in the second inequality if and only if the triangle degenerates into a straight line counted twice.

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.

A lampshade has the shape of a frustum of a right circular cone. Its perimeter is $P$ at the bottom, $p$ at the top, its slant height is $s$, and the inclination $i$ of the slant height to the bottom plane is $\geq 60°$; $P \geq p$. Show that such a lampshade can be cut out in one piece from a rectangular sheet of paper of dimensions $P$ and $s + \left[p(P-p)\right]/8s$. There will even be paper left over for a flap to glue the ends together, except in the extreme case where $P=p$, $i=\pi /2$, and no paper will be wasted.

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.