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Deviation measures and normals of convex bodies. (English) Zbl 1063.52003
For a convex body $$K$$ in $$E^n$$, denote by $$h_K(u)$$ its support function and by $$w_K(u) = h_K(u) + h_K(-u)$$ the width of $$K$$ in the direction $$u\in S^{n-1}$$. For a pair $$K, L$$ of such bodies, put $$\delta(K, L) = (\int_{S^{n-1}}(h_L(u)-h_K(u))^2dA)^{\frac 1 2}$$. The spherical deviation $$\mathcal S(K)$$ of $$K$$ is defined by $\mathcal S(K) = \inf \{\delta (K, L)\}$ over all balls $$L$$ in $$E^n$$. The eccentricity $$\mathcal E(K)$$ is defined by $\mathcal E(K) = \inf\{\delta (K, L)\}$ over all centrally symmetric bodies $$L$$ in $$E^n$$. Finally, the width deviation $$\mathcal W(K)$$ is given by $\mathcal W(K) = \frac1 2 \inf\left\{\left(\int_{S^{n-1}} (w_K(u)- w)^2dA\right)^{\frac 1 2} : w\geq 0\right\}.$ (There is a reason to avoid the family of all bodies of constant width for such a definition.)
Write $$\bar w(K)$$ for the mean width of $$K$$, i.e. put $$\bar w(K) =\frac {2}{\sigma_n}\int_{S^{n-1}}h_K(u)dA$$ where $$\sigma_n$$ is the area of $$S^{n-1}$$. Let $$B(K)$$ be the ball centered at the Steiner point of $$K$$ of radius $$\bar w(K)/2$$. Let $$K_*$$ be the central symmetrization of $$K$$ and $$K_0$$ the translate of $$K$$ that has its Steiner point at the origin.
The author proves (in Theorem 1) that
a) $$\mathcal S(K) =\delta (K,B(K))$$;
b) $$\mathcal E(K) = \delta (K_0,K_* )$$;
c) $$\mathcal W(K) = \delta (K_* ,B(K_* )) = \mathcal S(K_* )$$.
Moreover, $$\mathcal S(K)^2 = \mathcal E(K)^2 +\mathcal W(K)^2$$.
The relationship between the three deviation measures of a convex body $$K$$ and its normals (viewed as lines) has also been addressed (Section 4).
Note that, for a ball $$K$$, $$\mathcal S(K) = 0$$ and all the normals pass through the same point. For a body $$K$$ of constant width, $$\mathcal W(K) = 0$$ and all the normals are “double normals”. The author obtains an estimate for $$\mathcal S(K)$$ when normals of $$K$$ are close to a fixed point and an estimate for $$\mathcal W(K)$$ when any two parallel normals of $$K$$ are close to each other. A similar estimate is proved for $$\mathcal E(K)$$.
##### MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 52A27 Approximation by convex sets
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