He is a perfectly spherical boy.
Note that a beach ball isn't absolutely necessary - any spherical object will work as long as it's rigid and about the right size.
A convex body is said to have a curvature function, if its surface area measure is absolutely continuous with respect to spherical Lebesgue measure, and.
A convex body is said to have a -curvature function (see [2]), if its -surface area measure is absolutely continuous with respect to spherical Lebesgue measure, and.
A convex body is said to have an L p -curvature function [3]f p (K, ·): Sn-1→ ℝ, if its L p -surface area measure S p (K, ·) is absolutely continuous with respect to spherical Lebesgue measure S, and.
If the mixed surface area measure S i ( K, ⋅ ) is absolutely continuous with respect to spherical Lebesgue measure S, we have f p, i ( K, u ) = f i ( K, u ) h ( K, u ) 1 − p. (1.13).
A convex body K ∈ K 0 n is said to have a p-curvature function (see [21]) f p ( K, ⋅ ) : S n − 1 → R, if its L p surface area measure S p ( K, ⋅ ) is absolutely continuous with respect to spherical Lebesgue measure S and the Radon-Nikodym derivative d S p ( K, ⋅ ) d S = f p ( K, ⋅ ).
A convex body K ∈ K o n is said to have a L p -curvature function (see [14]) f p ( K, ⋅ ) : S n − 1 → R, if its L p surface area measure S p ( K, ⋅ ) is absolutely continuous with respect to spherical Lebesgue measure S and d S p ( K, ⋅ ) d S = f p ( K, ⋅ ).
If the i th surface area measure S i ( K, ⋅ ) is absolutely continuous with respect to the spherical Lebesgue measure S, we have f p, i ( K, ⋅ ) = h ( K, ⋅ ) 1 − p f i ( K, ⋅ ).
end{aligned} (2.18) If the ith surface area measure (S_{i}(K,cdot)) is absolutely continuous with respect to the spherical Lebesgue measure S, we have begin{aligned} f_{p,i}(K,cdot)=h K,cdot)^{1-p}f_{i} K, cdot).
Then, v | S R = g, i.e., series (27) converges (component-wise) uniformly and absolutely on the sphere S R. We shall prove that series (27) converges uniformly and absolutely in the spherical layer Σ R δ with arbitrarily small δ.
