Intrinsic volumes as averaged shadows
$$V_j(K)\;=\;\binom{n}{j}\frac{\kappa_n}{\kappa_j\,\kappa_{n-j}}\; \mathbb{E}_Q\big[\operatorname{Vol}_j(P_jQK)\big], \qquad n=3 .$$
A convex body under a random rotation, with its orthogonal shadow on a plane below
tetrahedron
cube
octahedron
dodecahedron
icosahedron
box, 2.8 × 1.8 × 0.7
irregular simplex
irregular polytope, 10 vertices
ball, radius 1
cylinder, r 0.8, h 2
disc, radius 1
segment, length 2
j = 2
j = 1
Sample
Run
×1000
Reset
Rotations drawn
0
Mean shadow area
—
Estimate of V
2
—
Exact V
2
—
Running estimate against the exact value (dashed)