*Article* **Volume Preserving Maps Between** *p***-Balls**

#### **Adrian Holho¸s \* and Daniela Ro¸sca**

Department of Mathematics, Technical University of Cluj-Napoca, str. Memorandumului 28, RO-400114 Cluj-Napoca, Romania; daniela.rosca@math.utcluj.ro

**\*** Correspondence: Adrian.Holhos@math.utcluj.ro

Received: 19 October 2019; Accepted: 12 November 2019; Published: 14 November 2019

**Abstract:** We construct a volume preserving map <sup>U</sup>*p* from the *p*-ball <sup>B</sup>*p*(*r*) = **x** ∈ R3, **<sup>x</sup>** *p* ≤ *r* to the regular octahedron B1(*<sup>r</sup>* ), for arbitrary *p* > 0. Then we calculate the inverse U −1 *p* and we also deduce explicit expressions for U∞ and U −1 ∞ . This allows us to construct volume preserving maps between arbitrary balls <sup>B</sup>*p*(*r*) and <sup>B</sup>*p* (*r*˜), and also to map uniform and refinable grids between them. Finally we list some possible applications of our maps.

**Keywords:** equal volume projection; hierarchical grid
