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Conditional constrained and unconstrained quantization for uniform distributions on regular polygons

Christina Hamilton, Evans Nyanney, Megha Pandey, Mrinal K. Roychowdhury

TL;DR

This work develops a unified framework for conditional constrained and unconstrained quantization of a uniform distribution on a regular polygon, with the vertex set as the conditioning baseline. Leveraging polygonal symmetry through a rotation map $T$, the authors derive exact structural forms for conditional optimal $n$-point sets when $n\ge k$, and obtain closed-form expressions for the distortion $V_n$, leading to the result that the conditional quantization dimension $D(P)=1$ and the conditional quantization coefficient is $\frac{2}{3} k^2 \sin^3(\pi/k)$. The paper then extends the analysis to constrained quantization under circumcircle, incircle, and diagonal constraints, providing explicit constructions and quantitative $V_n$ values, particularly for the hexagon. The results contribute a general, symmetry-based methodology for polygonal supports and offer concrete, computable guidance for applications where preassigned centers or geometric constraints influence optimal quantization, with potential relevance to resource placement and medical physics. Overall, the work establishes foundational theory and actionable examples for conditional and constrained quantization on polygonal domains.

Abstract

In this paper, we have considered a uniform distribution on a regular polygon with $k$-sides for some $k\geq 3$ and the set of all its $k$ vertices as a conditional set. For the uniform distribution under the conditional set first, for all positive integers $n\geq k$, we obtain the conditional optimal sets of $n$-points and the $n$th conditional quantization errors, and then we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Then, for the uniform distribution on the polygon taking the same conditional set, we investigate the conditional constrained optimal sets of $n$-points and the conditional constrained quantization errors for all $n \geq 6$, taking the constraint as the circumcircle, incircle, and then the different diagonals of the polygon.

Conditional constrained and unconstrained quantization for uniform distributions on regular polygons

TL;DR

This work develops a unified framework for conditional constrained and unconstrained quantization of a uniform distribution on a regular polygon, with the vertex set as the conditioning baseline. Leveraging polygonal symmetry through a rotation map , the authors derive exact structural forms for conditional optimal -point sets when , and obtain closed-form expressions for the distortion , leading to the result that the conditional quantization dimension and the conditional quantization coefficient is . The paper then extends the analysis to constrained quantization under circumcircle, incircle, and diagonal constraints, providing explicit constructions and quantitative values, particularly for the hexagon. The results contribute a general, symmetry-based methodology for polygonal supports and offer concrete, computable guidance for applications where preassigned centers or geometric constraints influence optimal quantization, with potential relevance to resource placement and medical physics. Overall, the work establishes foundational theory and actionable examples for conditional and constrained quantization on polygonal domains.

Abstract

In this paper, we have considered a uniform distribution on a regular polygon with -sides for some and the set of all its vertices as a conditional set. For the uniform distribution under the conditional set first, for all positive integers , we obtain the conditional optimal sets of -points and the th conditional quantization errors, and then we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Then, for the uniform distribution on the polygon taking the same conditional set, we investigate the conditional constrained optimal sets of -points and the conditional constrained quantization errors for all , taking the constraint as the circumcircle, incircle, and then the different diagonals of the polygon.
Paper Structure (9 sections, 23 theorems, 131 equations, 8 figures)

This paper contains 9 sections, 23 theorems, 131 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation and Significance.
  3. Preliminaries
  4. Conditional quantization for the uniform distribution $P$ on a regular $k$-gon for some $k\geq 3$
  5. Conditional constrained quantization for the uniform distribution $P$ on the regular $k$-gon when the constraint is the circumcircle
  6. Conditional constrained quantization for the uniform distribution $P$ on a regular $k$-gon when the constraint is the incircle
  7. Conditional constrained quantization for a uniform distribution $P$ on a regular polygon with $k$ sides when the constraint is a diagonal of the polygon
  8. When the constraint is a diagonal of the smaller length
  9. When the constraint is a diagonal of the larger length

Key Result

Proposition 2.1

(see PR3) Let $\alpha_n$ be an optimal set of $n$-points for $P$ such that $\alpha_n$ contains $m$ elements, for some $m\leq n$, from the side $L_1$ including the endpoints $A_1$ and $A_2$. Then, and the distortion error contributed by these $m$ elements from the side $L_1$ is given by

Figures (8)

  • Figure 1: The regular hexagon with the circumcircle and the incircle.
  • Figure 2: $A_1, A_2, A_3$ and $O$ form the conditional optimal set of four-points in an equilateral triangle.
  • Figure 3: In conditional quantization the dots represent the elements in an optimal set of $n$-points for $6\leq n\leq 13$.
  • Figure 4: In conditional constrained quantization with the constraint as the circumcircle, for a uniform distribution on a regular hexagon, the dots represent the elements in an optimal set of $n$-points for $6\leq n\leq 13$.
  • Figure 5: The regular hexagon with a circle inscribed in it.
  • ...and 3 more figures

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 53 more