Blaschke-Lebesgue-type Problem For Two Classes Of Convex Domains

This paper introduces the sum of width function DWn(Κ,θ) and use it to define two classes of convex domains Cn and Cn*, namely, Cn={Κ∈R2｜DWn(Κ,θ)= 4}, Cn*={Κ*｜Κ∈Cn}. Like the classical Blaschke-Lebesgue problem, this paper discusses the Blaschke-Lebesgue-type problem of Cn* and Cn. Combining the optimal control theory and mixed area method completely solves the Blaschke-Lebesgue-type problem of Cn*, obtaining all possible convex domains in C2m* and the curvilinear polygon R2m has the least area of all convex domains in C2m*. This paper prove that there exists only one element-unit disc-in C2m*-1 in two ways. On the problem of C2m-1, we get C2m-1= C1, then the Blaschke-Lebesgue-type problem of C2m-1 are the classical Blaschke-Lebesgue problem. On the problem of C2m, we formulated all convex domainsΚ's satisfyingλ(Κ)≥4(π-(?)-m tanπ/(4m)) through solving Blaschke-Lebesgue-type problem of C2m. Our estimates are better than Rogers-Shephard's.