Intrinsic volumes in dimension
4
$$V_j(K)\;=\;\binom{n}{j}\frac{\kappa_n}{\kappa_j\,\kappa_{n-j}}\; \mathbb{E}\big[\operatorname{Vol}_j(P_jK)\big], \qquad P_j \text{ onto a uniform random } j\text{-plane}.$$
The shadow of a convex body on a uniform random plane, with the previous shadows fading behind it
cube, side 2
graded box, sides 2i/n
random zonotope, n+2 generators
ball, radius 1
disc, radius 1
segment, length 2
2
3
4
5
6
7
8
9
10
11
12
j = 2
j = 1
Sample
Run
×1000
Reset
Planes drawn
0
Mean shadow area
—
Estimate of V
2
—
Exact V
2
—
Running estimate against the exact value (dashed)