Abstract. Some sharp bounds for the inner radius and the outer radius of the unit ball of a (normed or) Minkowski space with respect to its isoperimetrix are known. To find more such bounds is a challenging problem. Related to this motivation, we derive new sharp inequalities between inner and outer radii for the Holmes--Thompson and Busemann measures. Cross-section measures as well as the Blaschke--Santaló inequality will be used to obtain these new inequalities.

DOI: 10.14232/actasm-019-130-4

AMS Subject Classification (1991): 46B20, 52A20, 52A21, 52A40

Keyword(s): affine isoperimetric inequalities, Blaschke--Santalo inequality, Busemann measure, cross-section measures, Holmes--Thompson measure, inner radius, intersection body, isoperimetrix, Minkowski geometry, outer radius, projection body

received 30.6.2019, revised 23.9.2019, accepted 25.9.2019. (Registered under 630/2019.)