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Such lifts may yield compressed representations of polytopes, which are typically used to construct small-size linear programs. Motivated by algorithmic implications for the closest vector problem, we study lifts of Voronoi cells of lattices. We construct an explicit <jats:italic>d<\/jats:italic>-dimensional lattice such that every lift of the respective Voronoi cell has <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{\\Omega (d\/{\\log d})}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mrow>\n                      <mml:mi>\u03a9<\/mml:mi>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>\/<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>d<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> facets. On the positive side, we show that Voronoi cells of <jats:italic>d<\/jats:italic>-dimensional root lattices and their dual lattices have lifts with <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {O}}}(d)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {O}}}(d \\log d)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>log<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> facets, respectively. 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