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diff --git a/src/Coxeter_triangulation/include/gudhi/IO/build_mesh_from_cell_complex.h b/src/Coxeter_triangulation/include/gudhi/IO/build_mesh_from_cell_complex.h
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+/*    This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ *    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ *    Author(s):       Siargey Kachanovich
+ *
+ *    Copyright (C) 2019 Inria
+ *
+ *    Modification(s):
+ *      - YYYY/MM Author: Description of the modification
+ */
+
+#ifndef IO_BUILD_MESH_FROM_CELL_COMPLEX_H_
+#define IO_BUILD_MESH_FROM_CELL_COMPLEX_H_
+
+#include <gudhi/IO/output_debug_traces_to_html.h>  // for DEBUG_TRACES
+#include <gudhi/IO/Mesh_medit.h>
+
+#include <Eigen/Dense>
+
+#include <cstdlib>  // for std::size_t
+#include <map>
+#include <set>
+#include <string>
+#include <utility>    // for std::make_pair
+#include <algorithm>  // for std::min
+
+namespace Gudhi {
+
+namespace coxeter_triangulation {
+
+struct Configuration {
+  Configuration(bool t_edges, bool t_triangles, bool t_tetrahedra, std::size_t r_edges, std::size_t r_triangles,
+                std::size_t r_tetrahedra)
+      : toggle_edges(t_edges),
+        toggle_triangles(t_triangles),
+        toggle_tetrahedra(t_tetrahedra),
+        ref_edges(r_edges),
+        ref_triangles(r_triangles),
+        ref_tetrahedra(r_tetrahedra) {}
+
+  Configuration() {}
+
+  bool toggle_edges = true, toggle_triangles = true, toggle_tetrahedra = true;
+  std::size_t ref_edges = 1, ref_triangles = 1, ref_tetrahedra = 1;
+};
+
+template <class Hasse_cell, class Simplex_cell_map>
+void populate_mesh(Mesh_medit& output, Simplex_cell_map& sc_map, Configuration configuration, std::size_t amb_d,
+                   std::map<Hasse_cell*, std::size_t> vi_map) {
+  using Mesh_element_vertices = Mesh_medit::Mesh_elements::value_type::first_type;
+  std::map<Hasse_cell*, std::size_t> ci_map;
+  std::size_t index = vi_map.size() + 1;  // current size of output.vertex_points
+  if (sc_map.size() >= 3)
+    for (const auto& sc_pair : sc_map[2]) {
+      Eigen::VectorXd barycenter = Eigen::VectorXd::Zero(amb_d);
+      std::set<std::size_t> vertex_indices;
+      Hasse_cell* cell = sc_pair.second;
+      for (const auto& ei_pair : cell->get_boundary())
+        for (const auto& vi_pair : ei_pair.first->get_boundary()) vertex_indices.emplace(vi_map[vi_pair.first]);
+      for (const std::size_t& v : vertex_indices) barycenter += output.vertex_points[v - 1];
+      ci_map.emplace(cell, index++);
+      output.vertex_points.emplace_back((1. / vertex_indices.size()) * barycenter);
+#ifdef DEBUG_TRACES
+      std::string vlist = " (" + std::to_string(index - 1) + ")";
+      for (const std::size_t& v : vertex_indices) vlist += " " + std::to_string(v);
+      cell_vlist_map.emplace(to_string(cell), vlist);
+#endif
+    }
+
+  if (configuration.toggle_edges && sc_map.size() >= 2)
+    for (const auto& sc_pair : sc_map[1]) {
+      Hasse_cell* edge_cell = sc_pair.second;
+      Mesh_element_vertices edge;
+      for (const auto& vi_pair : edge_cell->get_boundary()) edge.push_back(vi_map[vi_pair.first]);
+      output.edges.emplace_back(edge, configuration.ref_edges);
+#ifdef DEBUG_TRACES
+      std::string vlist;
+      for (const std::size_t& v : edge) vlist += " " + std::to_string(v);
+      cell_vlist_map.emplace(to_string(edge_cell), vlist);
+#endif
+    }
+
+  if (configuration.toggle_triangles && sc_map.size() >= 3)
+    for (const auto& sc_pair : sc_map[2]) {
+      for (const auto& ei_pair : sc_pair.second->get_boundary()) {
+        Mesh_element_vertices triangle(1, ci_map[sc_pair.second]);
+        for (const auto& vi_pair : ei_pair.first->get_boundary()) triangle.push_back(vi_map[vi_pair.first]);
+        output.triangles.emplace_back(triangle, configuration.ref_triangles);
+      }
+    }
+
+  if (configuration.toggle_tetrahedra && sc_map.size() >= 4)
+    for (const auto& sc_pair : sc_map[3]) {
+      Eigen::VectorXd barycenter = Eigen::VectorXd::Zero(amb_d);
+      std::set<std::size_t> vertex_indices;
+      Hasse_cell* cell = sc_pair.second;
+      for (const auto& ci_pair : cell->get_boundary())
+        for (const auto& ei_pair : ci_pair.first->get_boundary())
+          for (const auto& vi_pair : ei_pair.first->get_boundary()) vertex_indices.emplace(vi_map[vi_pair.first]);
+      for (const std::size_t& v : vertex_indices) barycenter += output.vertex_points[v - 1];
+      output.vertex_points.emplace_back((1. / vertex_indices.size()) * barycenter);
+#ifdef DEBUG_TRACES
+      std::string vlist = " (" + std::to_string(index) + ")";
+      for (const std::size_t& v : vertex_indices) vlist += " " + std::to_string(v);
+      cell_vlist_map.emplace(to_string(cell), vlist);
+#endif
+
+      for (const auto& ci_pair : cell->get_boundary())
+        for (const auto& ei_pair : ci_pair.first->get_boundary()) {
+          Mesh_element_vertices tetrahedron = {index, ci_map[sc_pair.second]};
+          for (const auto& vi_pair : ei_pair.first->get_boundary()) tetrahedron.push_back(vi_map[vi_pair.first]);
+          output.tetrahedra.emplace_back(tetrahedron, configuration.ref_tetrahedra);
+        }
+      index++;
+    }
+}
+
+/** @brief Builds a Gudhi::coxeter_triangulation::Mesh_medit from a Gudhi::coxeter_triangulation::Cell_complex
+ * 
+ *  @ingroup coxeter_triangulation
+ */
+template <class Cell_complex>
+Mesh_medit build_mesh_from_cell_complex(const Cell_complex& cell_complex,
+                                        Configuration i_configuration = Configuration(),
+                                        Configuration b_configuration = Configuration()) {
+  using Hasse_cell = typename Cell_complex::Hasse_cell;
+  Mesh_medit output;
+  std::map<Hasse_cell*, std::size_t> vi_map;  // one for vertices, other for 2d-cells
+  std::size_t index = 1;                      // current size of output.vertex_points
+
+  if (cell_complex.cell_point_map().empty()) return output;
+  std::size_t amb_d = std::min((int)cell_complex.cell_point_map().begin()->second.size(), 3);
+
+  for (const auto& cp_pair : cell_complex.cell_point_map()) {
+#ifdef DEBUG_TRACES
+    std::string vlist;
+    vlist += " " + std::to_string(index);
+    cell_vlist_map.emplace(to_string(cp_pair.first), vlist);
+#endif
+    vi_map.emplace(cp_pair.first, index++);
+    output.vertex_points.push_back(cp_pair.second);
+    output.vertex_points.back().conservativeResize(amb_d);
+  }
+
+  populate_mesh(output, cell_complex.interior_simplex_cell_maps(), i_configuration, amb_d, vi_map);
+#ifdef DEBUG_TRACES
+  for (const auto& sc_map : cell_complex.interior_simplex_cell_maps())
+    for (const auto& sc_pair : sc_map) {
+      std::string simplex = "I" + to_string(sc_pair.first);
+      std::string cell = to_string(sc_pair.second);
+      std::string vlist = cell_vlist_map.at(cell).substr(1);
+      simplex_vlist_map.emplace(simplex, vlist);
+    }
+#endif
+  populate_mesh(output, cell_complex.boundary_simplex_cell_maps(), b_configuration, amb_d, vi_map);
+#ifdef DEBUG_TRACES
+  for (const auto& sc_map : cell_complex.boundary_simplex_cell_maps())
+    for (const auto& sc_pair : sc_map) {
+      std::string simplex = "B" + to_string(sc_pair.first);
+      std::string cell = to_string(sc_pair.second);
+      std::string vlist = cell_vlist_map.at(cell).substr(1);
+      simplex_vlist_map.emplace(simplex, vlist);
+    }
+#endif
+  return output;
+}
+
+}  // namespace coxeter_triangulation
+
+}  // namespace Gudhi
+
+#endif