Improvements of the isoperimetric inequality geometry of the Minkowski

V. I. Diskant

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003) no. 2, pp. 147-155

Résumé

The following improvements of an isoperimetric inequality in the $n$-dimensional Minkowski space $M^n$ ($n\geq 2$) with a normalizing body $B$ [3]: \begin{gather*} S^{\frac{n}{n-1}}_B(A) - (n^nV_B(I))^{\frac{1}{n-1}} V_B(A) \geq (S^{\frac{1}{n-1}}_B(A)-\rho(nV_B(I))^{\frac{1}{n-1}})^n -(n^nV_B(I))^{\frac{1}{n-1}}V_B(A_{-\rho }(I)), \\ S^{\frac{n}{n-1}}_B(A) - (n^nV_B(I_A))^{\frac{1}{n-1}} V_B(A) \geq (S^{\frac{1}{n-1}}_B(A)-\rho(nV_B(I_A))^{\frac{1}{n-1}})^n-(n^nV_B(I_A))^{\frac{1}{n-1}}V_B(A_{-\rho }(I)) \end{gather*} and series of their consequents, among which one improvement (11) of an isoperimetric inequality in $M^n$, taking into account both singularities on boundary of a body $A$, and deviation of body $A$ and $I_A$ from homothetic, improvement (13) an inequality of Hadwiger from [5] in $M^n$ in view of a nondegeneracy $A_{-q}(I)$, generalizing (15) of an inequality of Wills from [7] on $M^n$ are proved. In reduced inequalities $A$ — convex body, $I$ — isoperimetrix, $I_A$ — form-body of body $A$ relatively to $I$, $q$ — coefficient of holding capacity $I$ in $A$, $\rho\in [0,q]$, $A_{-\rho}(I)$ — internal body which is parallel to body $A$ relatively to $I$ on the distance $\rho$, $V_B(A)$ — the volume of body $A$, $S_B(A)$ — the surface area of body $A$ in $M^n$ [3].