9
votes
Accepted
Kinematic formula for Euler characteristic
Yes, this is called the principal kinematic formula:
$$\int \chi(K \cap gL)\, dg = \sum_{k=0}^n c_{nk} V_k(K) V_{n-k}(L),$$
where $V_i$ are the intrinsic volumes, and $c_{nk}$ certain constants. See e....
- 6,337
4
votes
Crofton formula: expected intersections is to length as variance is to what?
The formula you mentioned is true for a smooth curve $C$ on a unit sphere of any dimension and it is a special case of the Kac-Rice formula; see Example 15 here.
Let me explain a bit this point of ...
- 34.7k
2
votes
Accepted
Radon transform range theorem and radial functions
To answer Q1: There are trig identities at play. First, 0 is usually accepted under the definition of "homogeneous polynomial" (i.e., it's a polynomial whose coefficients are all zero) so ...
- 148
2
votes
Reconstructing a curve in $S^2$ from intersections with great circles
The answer to this question, as stated, is "no". If infinitely many intersections are allowed, it is easy to construct plenty of different curves, each of them having infinitely many intersections ...
- 91.8k
2
votes
Accepted
Reconstructing a curve in $S^2$ from intersections with great circles
This is related to the Funk transform. To a continuous function on the sphere it associates a map from the set of great circles to $\mathbb{R}$ that sends a great circle to the integral of the ...
- 6,337
1
vote
Reconstructing a curve in $S^2$ from intersections with great circles
You can't distinguish a curve $C$ from the curve $-C$.
- 26.3k
1
vote
Does this formula for caliper diameter hold for concave polyhedra?
No, this equation is false for non-convex polyhedra. Take a cube and remove from inside of it a smaller cube. The resulting body has the same mean width (caliper diameter), but the sum of angles times ...
- 6,337
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