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A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mi>C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is equal to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$2^{n+1}-1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and that the fractional illumination number of the polydisc in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mi>C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is equal to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$2^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.\n                  <\/jats:p>","DOI":"10.1007\/s00493-025-00195-7","type":"journal-article","created":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T11:53:35Z","timestamp":1767786815000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The complex Illumination problem"],"prefix":"10.1007","volume":"46","author":[{"given":"Liran","family":"Rotem","sequence":"first","affiliation":[]},{"given":"Alon","family":"Schejter","sequence":"additional","affiliation":[]},{"given":"Boaz A.","family":"Slomka","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,7]]},"reference":[{"issue":"11","key":"195_CR1","doi-asserted-by":"crossref","first-page":"3588","DOI":"10.1016\/j.jfa.2012.09.002","volume":"263","author":"J Abardia","year":"2012","unstructured":"Abardia, J.: Difference bodies in complex vector spaces. 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