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<p>In this paper, we establish a Blaschke-type theorem for path-connected and orthogonal convex sets in the plane. The separation of these sets is established using orthogonal convex paths generated by suitable grids. Consequently, a closed and orthogonal convex set can be represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonal convex sets in finite-dimensional spaces are also given. A procedure to find a staircase line that separates two disjoint, compact, path-connected, and orthogonal convex sets is shown. Numerical examples indicate that the separation of these sets by using the procedure is efficient.</p>

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Blaschke and Separation Theorems for Orthogonal Convex Sets and Some Applications

  • 期刊论文

  • 发表日期：

    2026年6月18日

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  • Nguyen Thi Le,
  • Phan Thanh An

摘要

In this paper, we establish a Blaschke-type theorem for path-connected and orthogonal convex sets in the plane. The separation of these sets is established using orthogonal convex paths generated by suitable grids. Consequently, a closed and orthogonal convex set can be represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonal convex sets in finite-dimensional spaces are also given. A procedure to find a staircase line that separates two disjoint, compact, path-connected, and orthogonal convex sets is shown. Numerical examples indicate that the separation of these sets by using the procedure is efficient.