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I guess you know that the result can be written as a polynomial. $$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$ So your question is how to estmate the coefficients $w_i$; they are this is so called "Cross-sectional measures", cross-sectional measures" and they can be defined for any convex body $K$.
• $w_0$ is the volume , of $K$,
• $w_n$ is the volume of unit ball.
• $w_1$ is the area of the boundary , of $K$, I think there is no good formula in generalfor ellipsoid, but estimates are known and you could write it as an integral.
• I am sure that there is no good formula for the rest of would be surprised if some remaining $w_i$.w_i$can be expressed by simple formula. If you want to write$w_i$as an integral, check for example Burago--Zalgaller, Geometric inequalities. Fore For example $$w_i=\mathrm{MayBeAConst}{\cdot}\int_{\partial w_i=\mathrm{MayBeAConst}\cdot\int\limits_{\partial K} \sigma_{n-i}(k_1,k_2,\dots,k_{n-1})\, sigma_{i-1}(k_1,k_2,\dots,k_{n-1})\, d\mathrm{area}.$$ where$\sigma_i$is the$i$-th elementary symmetric polynomial and$k_i$are principle curvatures. In your case it is easy to find$k_i$... 2 added 362 characters in body I guess you know that the result can be written as a polynomial. $$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$ So your question is how to estmate the coefficients .$w_i$; they are called "Cross-sectional measures", they can be defined for any convex body$K$. •$w_0$is the volume, •$w_n$is the volume of unit ball. •$w_1$is the area of the boundary, I think there is no good formula in general, but estimates are known and you could write it as an integral. • I am sure that there is no good formula for the rest of$w_i$. If you want to write$w_i$as an integral, check for example Burago--Zalgaller, Geometric inequalities. Fore example $$w_i=\mathrm{MayBeAConst}{\cdot}\int_{\partial K} \sigma_{n-i}(k_1,k_2,\dots,k_{n-1})\, d\mathrm{area}.$$ where$\sigma_i$is the$i$-th elementary symmetric polynomial and$k_i$are principle curvatures. In your case it is easy to find$k_i$... 1 I guess you know that the result can be written as a polynomial. $$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$ So your question is how to estmate the coefficients. •$w_0$is the volume, •$w_n$is the volume of unit ball. •$w_1$is the area of the boundary, I think there is no good formula in general, but estimates are known and you could write it as an integral. • I am sure that there is no good formula for the rest of$w_i$. If you want to write$w_i\$ as an integral, check for example Burago--Zalgaller, Geometric inequalities.