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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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